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<!doctype html> <html data-n-head-ssr lang="en" data-n-head="%7B%22lang%22:%7B%22ssr%22:%22en%22%7D%7D"> <head > <meta data-n-head="ssr" charset="utf-8"><meta data-n-head="ssr" name="viewport" content="width=device-width,initial-scale=1.0,maximum-scale=1.0, user-scalable=0"><meta data-n-head="ssr" http-equiv="Content-Security-Policy" content="default-src * data:; child-src * 'self' blob: http:;img-src * 'self' data: http:; script-src 'self' 'unsafe-inline' 'unsafe-eval' *;style-src 'self' 'unsafe-inline' *"><meta data-n-head="ssr" name="keywords" content="Mahalanobis distance,rail corrugation,evolution trend prediction,improved crested porcupine optimizer,hybrid time series network"><meta data-n-head="ssr" name="description" content="Analyzing the evolution trend of rail corrugation using signal processing and deep learning is critical for railway safety, as current traditional methods struggle to capture the complex evolution of corrugation. This present study addresses the challenge of accurately capturing this trend, which relies significantly on expert judgment, by proposing an intelligent prediction method based on self-attention (SA), a bidirectional temporal convolutional network (TCN), and a bidirectional gated recurrent unit (GRU). First, multidomain feature extraction and adaptive feature screening were used to obtain the optimal feature set. These features were then combined with principal component analysis (PCA) and the Mahalanobis distance (MD) method to construct a comprehensive health indicator (CHI) that reflects the evolution of rail corrugation. A bidirectional fusion model architecture was employed to capture the temporal correlations between forward and backward information during corrugation evolution, with SA embedded in the model to enhance the focus on key information. The outcome was a rail corrugation trend prediction network that combined a bidirectional TCN, bidirectional GRU, and SA. Subsequently, a multi-strategy improved crested porcupine optimizer (CPO) algorithm was constructed to automatically obtain the optimal network hyperparameters. The proposed method was validated with on-site rail corrugation data, demonstrating superior predictive performance compared to other advanced methods. In summary, the proposed method can accurately predict the evolution trend of rail corrugation, offering a valuable tool for on-site railway maintenance."><meta data-n-head="ssr" name="dc.title" content="Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU"><meta data-n-head="ssr" name="journal_id" content="ir.2024.20"><meta data-n-head="ssr" name="dc.date" content="2024-10-18"><meta data-n-head="ssr" name="dc.identifier" content="doi:10.20517/ir.2024.20"><meta data-n-head="ssr" name="dc.publisher" content="OAE Publishing Inc."><meta data-n-head="ssr" name="dc.type" content="Research Article"><meta data-n-head="ssr" name="dc.source" content=" Intell Robot 2024;4(4):318-38."><meta data-n-head="ssr" name="dc.citation.spage" content="318"><meta data-n-head="ssr" name="dc.citation.epage" content="338"><meta data-n-head="ssr" name="dc.creator" content="Jian-Hua Liu"><meta data-n-head="ssr" name="dc.creator" content="Wei-Hao Yang"><meta data-n-head="ssr" name="dc.creator" content="Jing He"><meta data-n-head="ssr" name="dc.creator" content="Zhong-Mei Wang"><meta data-n-head="ssr" name="dc.creator" content="Lin Jia"><meta data-n-head="ssr" name="dc.creator" content="Chang-Fan Zhang"><meta data-n-head="ssr" name="dc.creator" content="Wei-Wei Yang"><meta data-n-head="ssr" name="dc.subject" content="Mahalanobis distance"><meta data-n-head="ssr" name="dc.subject" content="rail corrugation"><meta data-n-head="ssr" name="dc.subject" content="evolution trend prediction"><meta data-n-head="ssr" name="dc.subject" content="improved crested porcupine optimizer"><meta data-n-head="ssr" name="dc.subject" content="hybrid time series network"><meta data-n-head="ssr" name="citation_reference" content="citation_title=Wang&nbsp;Z, Lei&nbsp;Z. 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Nonlin Dynam 2023;111:8419-38."><meta data-n-head="ssr" name="citation_journal_title" content="Intelligence & Robotics"><meta data-n-head="ssr" name="citation_publisher" content="OAE Publishing Inc."><meta data-n-head="ssr" name="citation_title" content="Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU"><meta data-n-head="ssr" name="citation_publication_date" content="2024/10/18"><meta data-n-head="ssr" name="citation_online_date" content="2024/10/18"><meta data-n-head="ssr" name="citation_doi" content="10.20517/ir.2024.20"><meta data-n-head="ssr" name="citation_volume" content="4"><meta data-n-head="ssr" name="citation_issue" content="4"><meta data-n-head="ssr" name="citation_firstpage" content="318"><meta data-n-head="ssr" name="citation_lastpage" content="338"><meta data-n-head="ssr" name="citation_author" content="Jian-Hua Liu"><meta data-n-head="ssr" name="citation_author" content="Wei-Hao Yang"><meta data-n-head="ssr" name="citation_author" content="Jing He"><meta data-n-head="ssr" name="citation_author" content="Zhong-Mei Wang"><meta data-n-head="ssr" name="citation_author" content="Lin Jia"><meta data-n-head="ssr" name="citation_author" content="Chang-Fan Zhang"><meta data-n-head="ssr" name="citation_author" content="Wei-Wei Yang"><meta data-n-head="ssr" name="prism.issn" content="ISSN 2770-3541 (Online)"><meta data-n-head="ssr" name="prism.publicationName" content="OAE Publishing Inc."><meta data-n-head="ssr" name="prism.publicationDate" content="2024-10-18"><meta data-n-head="ssr" name="prism.volume" content="4"><meta data-n-head="ssr" name="prism.section" content="Research Article"><meta data-n-head="ssr" name="prism.startingPag" content="318"><meta data-n-head="ssr" name="prism.url" content="https://www.oaepublish.com/articles/ir.2024.20"><meta data-n-head="ssr" name="prism.doi" content="doi:10.20517/ir.2024.20"><meta data-n-head="ssr" name="citation_journal_abbrev" content="ir"><meta data-n-head="ssr" name="citation_article_type" content="Research Article"><meta data-n-head="ssr" name="citation_language" content="en"><meta data-n-head="ssr" name="citation_doi" content="10.20517/ir.2024.20"><meta data-n-head="ssr" name="citation_id" content="ir.2024.20"><meta data-n-head="ssr" name="citation_issn" content="ISSN 2770-3541 (Online)"><meta data-n-head="ssr" name="citation_publication_date" content="2024-10-18"><meta data-n-head="ssr" name="citation_author_institution" content="Correspondence to: Dr. Zhong-Mei Wang, College of Railway Transportation, Hunan University of Technology, No 88, Taishan West Road, Taiyuan District, Zhuzhou 412007, Hunan, China. E-mail: wangzhongmei@hut.edu.cn"><meta data-n-head="ssr" name="citation_pdf_url" content="https://f.oaes.cc/xmlpdf/80d245c1-6593-4d42-8e24-7a9b32dfa757/ir4020.pdf"><meta data-n-head="ssr" name="citation_fulltext_html_url" content="https://www.oaepublish.com/articles/ir.2024.20"><meta data-n-head="ssr" name="fulltext_pdf" content="https://f.oaes.cc/xmlpdf/80d245c1-6593-4d42-8e24-7a9b32dfa757/ir4020.pdf"><meta data-n-head="ssr" name="twitter:type" content="article"><meta data-n-head="ssr" name="twitter:title" content="Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU"><meta data-n-head="ssr" name="twitter:description" content="Analyzing the evolution trend of rail corrugation using signal processing and deep learning is critical for railway safety, as current traditional methods struggle to capture the complex evolution of corrugation. This present study addresses the challenge of accurately capturing this trend, which relies significantly on expert judgment, by proposing an intelligent prediction method based on self-attention (SA), a bidirectional temporal convolutional network (TCN), and a bidirectional gated recurrent unit (GRU). First, multidomain feature extraction and adaptive feature screening were used to obtain the optimal feature set. These features were then combined with principal component analysis (PCA) and the Mahalanobis distance (MD) method to construct a comprehensive health indicator (CHI) that reflects the evolution of rail corrugation. A bidirectional fusion model architecture was employed to capture the temporal correlations between forward and backward information during corrugation evolution, with SA embedded in the model to enhance the focus on key information. The outcome was a rail corrugation trend prediction network that combined a bidirectional TCN, bidirectional GRU, and SA. Subsequently, a multi-strategy improved crested porcupine optimizer (CPO) algorithm was constructed to automatically obtain the optimal network hyperparameters. The proposed method was validated with on-site rail corrugation data, demonstrating superior predictive performance compared to other advanced methods. In summary, the proposed method can accurately predict the evolution trend of rail corrugation, offering a valuable tool for on-site railway maintenance."><meta data-n-head="ssr" name="og:url" content="https://www.oaepublish.com/articles/ir.2024.20"><meta data-n-head="ssr" name="og:type" content="article"><meta data-n-head="ssr" name="og:site_name" content="Intelligence & Robotics"><meta data-n-head="ssr" name="og:title" content="Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU"><meta data-n-head="ssr" name="og:description" content="Analyzing the evolution trend of rail corrugation using signal processing and deep learning is critical for railway safety, as current traditional methods struggle to capture the complex evolution of corrugation. This present study addresses the challenge of accurately capturing this trend, which relies significantly on expert judgment, by proposing an intelligent prediction method based on self-attention (SA), a bidirectional temporal convolutional network (TCN), and a bidirectional gated recurrent unit (GRU). First, multidomain feature extraction and adaptive feature screening were used to obtain the optimal feature set. These features were then combined with principal component analysis (PCA) and the Mahalanobis distance (MD) method to construct a comprehensive health indicator (CHI) that reflects the evolution of rail corrugation. A bidirectional fusion model architecture was employed to capture the temporal correlations between forward and backward information during corrugation evolution, with SA embedded in the model to enhance the focus on key information. The outcome was a rail corrugation trend prediction network that combined a bidirectional TCN, bidirectional GRU, and SA. Subsequently, a multi-strategy improved crested porcupine optimizer (CPO) algorithm was constructed to automatically obtain the optimal network hyperparameters. The proposed method was validated with on-site rail corrugation data, demonstrating superior predictive performance compared to other advanced methods. In summary, the proposed method can accurately predict the evolution trend of rail corrugation, offering a valuable tool for on-site railway maintenance."><title>Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU</title><link data-n-head="ssr" rel="icon" type="image/x-icon" href="/favicon.ico"><link data-n-head="ssr" rel="canonical" href="https://www.oaepublish.com/articles/ir.2024.20"><script data-n-head="ssr" src="https://accounts.google.com/gsi/client" async></script><script data-n-head="ssr" src="https://g.oaes.cc/oae/dist/relijs.js" async></script><script data-n-head="ssr" src="https://www.googletagmanager.com/gtag/js?id=G-FM6KBJGRBV" async></script><link rel="preload" href="https://g.oaes.cc/oae/nuxt/b06ddfb.js" as="script"><link rel="preload" href="https://g.oaes.cc/oae/nuxt/0a3b980.js" as="script"><link rel="preload" href="https://g.oaes.cc/oae/nuxt/css/8176b15.css" as="style"><link rel="preload" href="https://g.oaes.cc/oae/nuxt/3e8004d.js" as="script"><link rel="preload" href="https://g.oaes.cc/oae/nuxt/css/f3a19d3.css" as="style"><link rel="preload" href="https://g.oaes.cc/oae/nuxt/b19d7ea.js" as="script"><link rel="preload" href="https://g.oaes.cc/oae/nuxt/css/52ac674.css" as="style"><link rel="preload" href="https://g.oaes.cc/oae/nuxt/d1923ac.js" as="script"><link rel="stylesheet" href="https://g.oaes.cc/oae/nuxt/css/8176b15.css"><link rel="stylesheet" href="https://g.oaes.cc/oae/nuxt/css/f3a19d3.css"><link rel="stylesheet" href="https://g.oaes.cc/oae/nuxt/css/52ac674.css"> </head> <body > <div data-server-rendered="true" id="__nuxt"><div id="__layout"><div data-fetch-key="data-v-0baa1603:0" data-v-0baa1603><div class="PcComment" data-v-43aad25a data-v-0baa1603><div class="ipad_bg" style="display:none;" data-v-43aad25a></div> <div class="head_top" data-v-43aad25a><div class="wrapper head_box" data-v-43aad25a><span class="qk_jx" data-v-43aad25a><img src="https://i.oaes.cc/upload/journal_logo/ir.png" alt data-v-43aad25a></span> <a href="/ir" class="qk_a_name" data-v-43aad25a><span class="title font20" data-v-43aad25a>Intelligence & Robotics</span></a> <i class="el-icon-caret-right sjbtn" style="color:rgb(0,71,187);" data-v-43aad25a></i> <div class="top_img" data-v-43aad25a><a href="https://www.scopus.com/sourceid/21101199351" target="_blank" data-v-43aad25a><img src="https://i.oaes.cc/uploads/20240813/49390c7e86ab40a58ee862e8c1af65ba.png" alt data-v-43aad25a></a><a href="" target="_blank" data-v-43aad25a><img src="https://i.oaes.cc/uploads/20240506/ea3d9071c35b4bf3982ffe25f1083620.png" alt data-v-43aad25a></a></div> <div class="oae_menu_box" data-v-43aad25a><a href="/alljournals" data-v-43aad25a><span data-v-43aad25a>All Journals</span></a></div> <span class="search" data-v-43aad25a><i class="icon-search icon_right font24" data-v-43aad25a></i> <span data-v-43aad25a>Search</span></span> <span class="go_oae" data-v-43aad25a><a href="https://oaemesas.com/login?JournalId=ir" target="_blank" data-v-43aad25a><i class="icon-login-line icon_right font24" data-v-43aad25a></i> <span data-v-43aad25a>Log In</span></a></span></div></div> <div class="cg" style="height: 41px" data-v-43aad25a></div>  <div class="head_text" style="border-bottom:3px solid rgb(0,71,187);" data-v-43aad25a><div class="head_search wrapper" style="display:none;" data-v-43aad25a><div class="box_btn" data-v-43aad25a><div class="qk_miss" data-v-43aad25a><img src="https://i.oaes.cc/uploads/20231121/59802903b17e4eebae240e004311d193.jpg" alt class="qk_fm" data-v-43aad25a> <div class="miss_right" data-v-43aad25a><div class="miss_btn" data-v-43aad25a><span data-v-43aad25a><span class="font_b" data-v-43aad25a>Editor-in-Chief:</span> Simon X. 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class="tit_box mgt30" data-v-6dffe839><h1 id="art_title" class="art_title2" data-v-6dffe839><span data-v-6dffe839>Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU</span></h1> <div class="art_seltte" data-v-6dffe839><i class="iconfont icon-yuyan" data-v-6dffe839></i> <div class="el-select selete_language" data-v-6dffe839><div class="el-input el-input--suffix"><input type="text" readonly="readonly" autocomplete="off" placeholder="" class="el-input__inner"><span class="el-input__suffix"><span class="el-input__suffix-inner"><i class="el-select__caret el-input__icon el-icon-arrow-up"></i></span></span></div><div class="el-select-dropdown el-popper" style="min-width:;display:none;"><div class="el-scrollbar" style="display:none;"><div class="el-select-dropdown__wrap el-scrollbar__wrap el-scrollbar__wrap--hidden-default"><ul class="el-scrollbar__view el-select-dropdown__list"><li class="el-select-dropdown__item" 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class="Crossref" data-v-6dffe839> <span data-v-6dffe839>0</span> </span></div> <div id=" authorString" class="article-authors" data-v-6dffe839><span class="authors_item" data-v-6dffe839><div affNumList="" data-v-dc220f24 data-v-6dffe839><span class="pos_re" data-v-dc220f24><div role="tooltip" id="el-popover-601" aria-hidden="true" class="el-popover el-popper" style="width:300px;display:none;"><h3 class="font16 no_sup" style="color:#333;margin-bottom:20px;" data-v-dc220f24>Jian-Hua Liu<sup>1</sup></h3> <div class="Aff_current font14 no_sup" data-v-dc220f24><div data-v-dc220f24><div class="author_cont" data-v-dc220f24><label>1</label><addr-line>College of Railway Transportation, Hunan University of Technology, Zhuzhou 412007, Hunan, China.</addr-line></div></div><div data-v-dc220f24><div class="author_cont" data-v-dc220f24><label>2</label><addr-line>College of Electrical and Information Engineering, Hunan University of Technology, Zhuzhou 412007, Hunan, China.</addr-line></div></div><div data-v-dc220f24><div class="author_cont" data-v-dc220f24><label>3</label><addr-line>Zhuzhou Qingyun Electric Locomotive Accessories Factory Co., Ltd, Zhuzhou 412007, Hunan, China.</addr-line></div></div></div> <i class="close_btn el-icon-close" data-v-dc220f24></i> <a href="https://scholar.google.com/scholar?q=Jian-Hua Liu" target="_blank" data-v-dc220f24><button type="button" class="el-button el-button--primary el-button--mini" data-v-dc220f24><span>Google Scholar</span></button></a></div><span class="el-popover__reference-wrapper"><span class="author_name" data-v-dc220f24>Jian-Hua Liu<sup>1</sup></span></span></span></div>     <i data-v-6dffe839> , </i></span><span class="authors_item" data-v-6dffe839><div affNumList="" data-v-dc220f24 data-v-6dffe839><span class="pos_re" data-v-dc220f24><div role="tooltip" id="el-popover-3698" aria-hidden="true" class="el-popover el-popper" style="width:300px;display:none;"><h3 class="font16 no_sup" style="color:#333;margin-bottom:20px;" data-v-dc220f24>Wei-Hao Yang<sup>1</sup></h3> <div class="Aff_current font14 no_sup" data-v-dc220f24><div data-v-dc220f24><div class="author_cont" data-v-dc220f24><label>1</label><addr-line>College of Railway Transportation, Hunan University of Technology, Zhuzhou 412007, Hunan, China.</addr-line></div></div><div data-v-dc220f24><div class="author_cont" data-v-dc220f24><label>2</label><addr-line>College of Electrical and Information Engineering, Hunan University of Technology, Zhuzhou 412007, Hunan, China.</addr-line></div></div><div data-v-dc220f24><div class="author_cont" data-v-dc220f24><label>3</label><addr-line>Zhuzhou Qingyun Electric Locomotive Accessories Factory Co., Ltd, Zhuzhou 412007, Hunan, China.</addr-line></div></div></div> <i class="close_btn el-icon-close" data-v-dc220f24></i> <a href="https://scholar.google.com/scholar?q=Wei-Hao Yang" target="_blank" data-v-dc220f24><button type="button" class="el-button el-button--primary el-button--mini" 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University of Technology, Zhuzhou 412007, Hunan, China.</addr-line></div></div><div data-v-dc220f24><div class="author_cont" data-v-dc220f24><label>2</label><addr-line>College of Electrical and Information Engineering, Hunan University of Technology, Zhuzhou 412007, Hunan, China.</addr-line></div></div><div data-v-dc220f24><div class="author_cont" data-v-dc220f24><label>3</label><addr-line>Zhuzhou Qingyun Electric Locomotive Accessories Factory Co., Ltd, Zhuzhou 412007, Hunan, China.</addr-line></div></div></div> <i class="close_btn el-icon-close" data-v-dc220f24></i> <a href="https://scholar.google.com/scholar?q=Wei-Wei Yang" target="_blank" data-v-dc220f24><button type="button" class="el-button el-button--primary el-button--mini" data-v-dc220f24><span>Google Scholar</span></button></a></div><span class="el-popover__reference-wrapper"><span class="author_name" data-v-dc220f24>Wei-Wei Yang<sup>3</sup></span></span></span></div>     </span> <button type="button" class="el-button el-button--primary el-button--mini" data-v-6dffe839><i class="el-icon-plus"></i><span>Show Authors</span></button></div> <div class="article-header-info" data-v-6dffe839><div data-v-6dffe839> <i>Intell Robot</i> 2024;4(4):318-38.</div> <div class="mgt5" data-v-6dffe839><a href="https://doi.org/10.20517/ir.2024.20" target="_blank" data-v-6dffe839>10.20517/ir.2024.20</a> | <span class="btn_link" data-v-6dffe839>© The Author(s) 2024.</span></div></div> <div class="top_btn_box" data-v-6dffe839><div class="btn_item" data-v-6dffe839><i class="el-icon-caret-right" data-v-6dffe839></i><span data-v-6dffe839>Author Information</span></div> <div class="btn_item" data-v-6dffe839><i class="el-icon-caret-right" data-v-6dffe839></i><span data-v-6dffe839>Article Notes</span></div> <div class="btn_item" data-v-6dffe839><i class="el-icon-caret-right" data-v-6dffe839></i><span data-v-6dffe839>Cite This Article</span></div></div> <div class="author_box" style="display:none;" data-v-6dffe839><div data-v-6dffe839><div data-v-6dffe839><label>1</label><addr-line>College of Railway Transportation, Hunan University of Technology, Zhuzhou 412007, Hunan, China.</addr-line></div></div><div data-v-6dffe839><div data-v-6dffe839><label>2</label><addr-line>College of Electrical and Information Engineering, Hunan University of Technology, Zhuzhou 412007, Hunan, China.</addr-line></div></div><div data-v-6dffe839><div data-v-6dffe839><label>3</label><addr-line>Zhuzhou Qingyun Electric Locomotive Accessories Factory Co., Ltd, Zhuzhou 412007, Hunan, China.</addr-line></div></div> <div class="CorrsPlus" data-v-6dffe839><div data-v-6dffe839><span id="cirrsMail" data-v-6dffe839>Correspondence to: Dr. Zhong-Mei Wang, College of Railway Transportation, Hunan University of Technology, No 88, Taishan West Road, Taiyuan District, Zhuzhou 412007, Hunan, China. E-mail: <email>wangzhongmei@hut.edu.cn</email></span></div></div></div> <div class="notes_box" style="display:none;" data-v-6dffe839><div class="articleDate mag_top10" data-v-6dffe839><span><b>Received:</b> 28 Aug 2024 | </span><span><b>First Decision:</b> 14 Sep 2024 | </span><span><b>Revised:</b> 21 Sep 2024 | </span><span><b>Accepted:</b> 11 Oct 2024 | </span><span><b>Published:</b> 18 Oct 2024</span></div> <div class="articleDate" data-v-6dffe839><span><b>Academic Editor:</b> Simon Yang | </span><span><b>Copy Editor:</b> Pei-Yun Wang | </span><span><b>Production Editor:</b> Pei-Yun Wang</span></div></div> <div class="article_bg" data-v-6dffe839><h2 id="art_Abstract" data-v-6dffe839>Abstract</h2> <div id="seo_des" class="article_Abstract mag_btn10" data-v-6dffe839><p>Analyzing the evolution trend of rail corrugation using signal processing and deep learning is critical for railway safety, as current traditional methods struggle to capture the complex evolution of corrugation. This present study addresses the challenge of accurately capturing this trend, which relies significantly on expert judgment, by proposing an intelligent prediction method based on self-attention (SA), a bidirectional temporal convolutional network (TCN), and a bidirectional gated recurrent unit (GRU). First, multidomain feature extraction and adaptive feature screening were used to obtain the optimal feature set. These features were then combined with principal component analysis (PCA) and the Mahalanobis distance (MD) method to construct a comprehensive health indicator (CHI) that reflects the evolution of rail corrugation. A bidirectional fusion model architecture was employed to capture the temporal correlations between forward and backward information during corrugation evolution, with SA embedded in the model to enhance the focus on key information. The outcome was a rail corrugation trend prediction network that combined a bidirectional TCN, bidirectional GRU, and SA. Subsequently, a multi-strategy improved crested porcupine optimizer (CPO) algorithm was constructed to automatically obtain the optimal network hyperparameters. The proposed method was validated with on-site rail corrugation data, demonstrating superior predictive performance compared to other advanced methods. In summary, the proposed method can accurately predict the evolution trend of rail corrugation, offering a valuable tool for on-site railway maintenance.</p></div>  <h2 id="art_Keywords" data-v-6dffe839>Keywords</h2> <div class="article_Abstract" data-v-6dffe839><span data-v-6dffe839><span data-v-6dffe839>Mahalanobis distance</span><i data-v-6dffe839>, </i></span><span data-v-6dffe839><span data-v-6dffe839>rail corrugation</span><i data-v-6dffe839>, </i></span><span data-v-6dffe839><span data-v-6dffe839>evolution trend prediction</span><i data-v-6dffe839>, </i></span><span data-v-6dffe839><span data-v-6dffe839>improved crested porcupine optimizer</span><i data-v-6dffe839>, </i></span><span data-v-6dffe839><span data-v-6dffe839>hybrid time series network</span></span></div></div> <div class="MoComment" data-v-6dffe839><div class="top_banner" data-v-6dffe839><div class="oae_header" data-v-6dffe839>Author's Talk</div> <div class="line" data-v-6dffe839></div> <div class="img_box" data-v-6dffe839><img src="https://i.oaes.cc/uploads/20241021/d5f21a891ccd43739a59e3f817c67e42.png" alt="" data-itemid="7277" data-itemhref="https://v.oaes.cc/uploads/20241021/6879d4f4c749472a807548446dea5a90.mp4" data-itemimg="https://i.oaes.cc/uploads/20241021/d5f21a891ccd43739a59e3f817c67e42.png" data-v-6dffe839> <i data-itemid="7277" data-itemhref="https://v.oaes.cc/uploads/20241021/6879d4f4c749472a807548446dea5a90.mp4" data-itemimg="https://i.oaes.cc/uploads/20241021/d5f21a891ccd43739a59e3f817c67e42.png" class="bo_icon" data-v-6dffe839></i></div></div>  <div class="article_link" data-v-6dffe839><span data-v-6dffe839><a href="https://f.oaes.cc/xmlpdf/80d245c1-6593-4d42-8e24-7a9b32dfa757/ir4020_down.pdf?v=65" data-v-6dffe839><b data-v-6dffe839><i class="icon-download icon_right4" data-v-6dffe839></i> Download PDF</b></a></span> <span data-v-6dffe839><i class="comment-l icon-commentl iconfont icon_right4" data-v-6dffe839></i> <b data-v-6dffe839>0</b></span> <span data-v-6dffe839><span data-v-6dffe839><div role="tooltip" id="el-popover-796" aria-hidden="true" class="el-popover el-popper" style="width:170px;display:none;"><div class="icon_share" style="text-align:right;margin:0;" data-v-6dffe839><a href="http://pinterest.com/pin/create/button/?url=&media=&description=https://www.oaepublish.com/articles/" target="_blank" class="pinterest-sign" data-v-6dffe839><i class="iconfont icon-pinterest" data-v-6dffe839></i></a> <a href="https://www.facebook.com/sharer/sharer.php?u=https://www.oaepublish.com/articles/" target="_blank" class="facebook-sign" data-v-6dffe839><i aria-hidden="true" class="iconfont icon-facebook" data-v-6dffe839></i></a> <a href="https://twitter.com/intent/tweet?url=https://www.oaepublish.com/articles/" target="_blank" class="twitter-sign" data-v-6dffe839><i class="iconfont icon-tuite1" data-v-6dffe839></i></a> <a href="https://www.linkedin.com/shareArticle?url=https://www.oaepublish.com/articles/" target="_blank" class="linkedin-sign" data-v-6dffe839><i class="iconfont icon-linkedin" data-v-6dffe839></i></a></div> </div><span class="el-popover__reference-wrapper"><button type="button" class="el-button colorddd el-button--text el-button--mini" data-v-6dffe839><span><i class="icon-fenxiang iconfont icon_right4" data-v-6dffe839></i> <b data-v-6dffe839>3</b></span></button></span></span></span> <span data-v-6dffe839><span class="no_zan" data-v-6dffe839><i class="icon-like-line icon_right4" data-v-6dffe839></i> <i class="num_n" data-v-6dffe839><b data-v-6dffe839>10</b></i></span></span></div></div> <div id="artDivBox" class="art_cont" data-v-6dffe839><div id="s1" class="article-Section"><h2 >1. INTRODUCTION</h2><p class="">Long-term wheel-rail contact on railway lines can cause various types of damage, particularly in sections with small curvature radius, where corrugation damage is more prevalent<sup>[<a href="#b1" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b1">1</a>-<a href="#b3" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b3">3</a>]</sup>. Rail corrugation primarily affects the inner surface of the rail in a curved section, resulting in periodic wavy wear. If left undetected and unrepaired, corrugation can cause train vibrations, significantly reducing its operating stability. In severe cases, rail breakage and major accidents, such as train derailments, can also occur<sup>[<a href="#b4" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b4">4</a>,<a href="#b5" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b5">5</a>]</sup>. Therefore, in railway health management, in-depth research on the evolution of rail corrugation is critical<sup>[<a href="#b6" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b6">6</a>]</sup> to ensuring the safe operation of rail transit<sup>[<a href="#b7" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b7">7</a>]</sup>.</p><p class="">Over the past years, scholars have conducted in-depth research on the generation and evolution process of rail corrugation, mainly using two methods: mechanism modeling and data-driven prediction. In mechanism modeling, the wheel-rail transient dynamics method is used to establish a model that reflects the evolution process of corrugation. Additionally, by using mechanical simulation software, scholars have constructed wheel-rail coupling finite element models and rail elastic-plastic analysis models to further explore the generation and evolution<sup>[<a href="#b8" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b8">8</a>,<a href="#b9" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b9">9</a>]</sup> of corrugation. For example, Wang <i>et al</i>. established a vehicle-track space coupling model using multibody dynamics software and conducted a dynamic analysis of the corrugation section<sup>[<a href="#b1" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b1">1</a>]</sup>. Cui <i>et al</i>. established a finite element model of the wheel-rail system and a wear model for corrugation using typical rail corrugation on a curve with a small radius as the research object; they then elucidated the development mechanism of corrugation by studying the dynamic response of the wheel-rail on the rail surface<sup>[<a href="#b2" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b2">2</a>]</sup>. However, these methods rely on prior knowledge of factors, such as the damage mechanism, and are highly theoretical. Furthermore, achieving an optimal damage evolution process using these methods in a complex train operating environment is challenging.</p><p class="">In data-driven research, scholars typically use experimental or on-site data to extract damage degradation features. Machine and deep learning methods are employed for damage diagnosis or prediction tasks without requiring an in-depth understanding of the internal damage mechanisms, as these methods can indirectly consider various influencing factors<sup>[<a href="#b10" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b10">10</a>-<a href="#b13" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b13">13</a>]</sup>. For example, Xiao <i>et al</i>. used machine learning to detect and assess corrugation damage in heavy haul railways; their approach, which was based on support vector machines and other technologies, could effectively detect rail corrugation damage<sup>[<a href="#b14" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b14">14</a>]</sup>. Deep network models such as gated recurrent units (GRUs), temporal convolutional networks (TCNs), and attention mechanisms are widely used in industrial equipment for damage diagnosis and degradation trend prediction<sup>[<a href="#b15" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b15">15</a>-<a href="#b20" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b20">20</a>]</sup> because of their exceptional feature extraction and nonlinear mapping abilities. For example, Zhang <i>et al</i>. introduced a squeeze-excitation channel attention mechanism into a combined model of a convolutional neural network (CNN) and bidirectional GRU (BiGRU); this integration demonstrated that the addition of an attention mechanism improved the capability of the network to focus on excellent features<sup>[<a href="#b21" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b21">21</a>]</sup>. Liu <i>et al</i>. used a dynamic multiscale gated causal convolution method combined with a GRU to effectively predict the actual degradation trend of rail corrugation and address poor generalization caused by small data samples<sup>[<a href="#b22" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b22">22</a>]</sup>. Additionally, in the general damage evolution prediction task, degradation is a continuous change process with a front-back relationship over time<sup>[<a href="#b23" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b23">23</a>]</sup>. Currently, most scholars do not consider the relationship between the time series before and after the damage signal. In a complex time-series prediction task, a single model often has limitations in terms of generalization, robustness, and adaptability. The current hybrid temporal prediction networks typically rely on extensive experiments and parameter-tuning processes, thereby increasing the computational cost and making the optimality of the selected hyperparameters difficult.</p><p class="">To address the aforementioned shortcomings, this study constructed a self-attention (SA) bidirectional TCN and GRU (SA-BiTCN-BiGRU) hybrid network and used a new multi-strategy improved crested porcupine optimizer (MICPO) algorithm for automatic hyperparameter optimization. The proposed model integrated the advantages of each module, exhibiting robust time-series modeling capabilities, perceiving dynamic changes in a time series, and assigning more weight to important time-series features. Thus, the prediction accuracy of the evolution trend of rail corrugation improved. The MICPO algorithm could automatically determine the optimal network hyperparameters for the proposed network using the four improvement strategies and its superior global search ability, thereby enhancing the network's prediction accuracy and reducing the need for blind manual adjustment of hyperparameters. Finally, the efficacy and superiority of the proposed methodology were verified through experiments and compared with other advanced methods.</p><p class="">The remainder of this study is organized as follows. Section 2 introduces the construction method of the rail corrugation's comprehensive health indicator (CHI), corrugation evolution trend prediction model, and model hyperparameter optimization algorithm. Section 3 describes the experimental setup and preprocessing of the rail corrugation dataset, and subsequently analyzes the experimental results in detail. Section 4 provides a comprehensive summary of the research content and proposes current limitations and future research directions. Finally, Section 5 concludes the study.</p></div><div id="s2" class="article-Section"><h2 >2. METHODS</h2><p class="">Based on the current research background, this section provides a detailed description of the process for predicting the evolutionary trend of rail corrugation. A corrugation CHI was established using the collected on-site dataset. A SA-BiTCN-BiGRU hybrid network was used to predict the evolution trend of rail corrugation, and the MICPO algorithm was constructed to adaptively adjust the hyperparameters of the network. The overall framework is illustrated in <a href="#Figure1" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure1">Figure 1</a>.</p><div class="Figure-block" id="Figure1"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2024.20/image/Figure1" class="Article-img" alt="" target="_blank"><img alt="Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU" src="https://image.oaes.cc/80d245c1-6593-4d42-8e24-7a9b32dfa757/ir4020-1.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 1. Frame diagram of rail corrugation trend prediction.</p></div></div><p class="">As can be seen from <a href="#Figure1" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure1">Figure 1</a>, first, by observing the damage changes in the corrugation image, the three vibration sensors were installed at the front wheel, rear wheel, and center position of the bogie on the track inspection car to collect corrugation vibration data in the vertical direction. After preprocessing the collected data, the rail corrugation vibration signal was obtained. Subsequently, multidomain feature extraction, feature screening, feature dimensionality reduction, and the Mahalanobis distance (MD) measurement methods were applied to this corrugation vibration signal, resulting in a CHI that effectively characterized the evolution trend of rail corrugation. The CHI was then input into the SA-BiTCN-BiGRU hybrid network to predict the evolution trend of rail corrugation. The network integrated the advantages of BiTCN, BiGRU, and SA to address the limitations of existing models. Finally, the MICPO algorithm was used to accurately select the optimal network model hyperparameters, thereby effectively improving the prediction accuracy of the model.</p><div id="s2-1" class="article-Section"><h3 >2.1. Collection of rail corrugation signal and construction of corrugation CHI</h3><p class="">In this study, three vibration sensors installed on the track inspection car were used to obtain vibration data of corrugation damage from different positions in the same direction. Compared with the data of a single sensor, the multi-channel data contains richer feature information and can more comprehensively reflect the changing characteristics of corrugation damage<sup>[<a href="#b24" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b24">24</a>]</sup>. Therefore, to fully explore the vibration information of the three channels, we first normalized the data of each channel to reduce the impact of the difference in signal distribution between different channels. The multi-channel signal fusion method based on kurtosis weight was then used to calculate the fusion weight of the three channels, and the signals from each channel were subjected to weighted fusion. The kurtosis value can effectively reflect the severity of rail corrugation damage. The channel with a higher kurtosis value is considered to be more sensitive to the reflection of corrugation damage, so a higher weight is assigned to ensure that more representative vibration signals have a more significant impact on the overall analysis results during the fusion process, so that the merged vibration signals can reflect the changing trend of rail corrugation damage more comprehensively and reliably<sup>[<a href="#b25" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b25">25</a>,<a href="#b26" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b26">26</a>]</sup>.</p><p class="">The CHI is an indicator used to evaluate and quantify the evolution trend of rail corrugation. The construction of a CHI is a preprocessing step for predicting the evolution of corrugation, which influences the effectiveness of subsequent prediction tasks<sup>[<a href="#b27" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b27">27</a>]</sup>. However, in a complex environment, various adverse factors may lead to significant deviations in the extracted rail corrugation vibration data, resulting in a lack of reliability in the constructed CHI. Therefore, this study used a custom range box line method to identify outliers in the corrugation vibration data and performed a mean correction on these outliers, thereby improving data quality. To accurately construct the CHI of corrugation and overcome the problem of relying on manual experience selection for single physical and fusion indicators, this study establishes a CHI that reflects the evolution of corrugation. The process steps are described as follows.</p><p class="">First, the vibration data of rail corrugation collected from the field contain numerous degradation features reflecting the evolution process of corrugation. The amplitude of these features usually deviates from the normal range with time, indicating that the corrugation damage is intensifying<sup>[<a href="#b28" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b28">28</a>]</sup>. Therefore, this study extracted time-domain, frequency-domain, and time-frequency domain feature indicators from the data, such as the maximum value, root mean square (RMS), standard deviation, and pulse index. These feature indicators effectively reflect corrugation degradation through the concretization of abstract real data.</p><p class="">Subsequently, three evaluation indicators, monotonicity (<inline-formula><tex-math id="M1">$$ M $$</tex-math></inline-formula>), Spearman's correlation coefficient (<inline-formula><tex-math id="M2">$$ S $$</tex-math></inline-formula>), and robustness (<inline-formula><tex-math id="M3">$$ R $$</tex-math></inline-formula>), were used to quantify the damage features of rail corrugation<sup>[<a href="#b29" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b29">29</a>]</sup>. Concurrently, to conduct comprehensive evaluation of each rail's corrugation features, this study used normalization processing to quantify the three evaluation indicators to the same scale, eliminating the impact of dimension, and obtained comprehensive evaluation indicator (<inline-formula><tex-math id="M4">$$ C $$</tex-math></inline-formula>)<sup>[<a href="#b30" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b30">30</a>,<a href="#b31" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b31">31</a>]</sup> through linear weighted combination of the three evaluation indicators, using it to calculate the comprehensive score of each feature, and adaptively screen out the features sensitive to the change of rail's corrugation state, forming an optimal feature subset. The definitions of <inline-formula><tex-math id="M5">$$ M $$</tex-math></inline-formula>, <inline-formula><tex-math id="M6">$$ S $$</tex-math></inline-formula>, <inline-formula><tex-math id="M7">$$ R $$</tex-math></inline-formula>, and <inline-formula><tex-math id="M8">$$ C $$</tex-math></inline-formula> are</p><p class=""><div class="disp-formula"><label>(1)</label><tex-math id="E1"> $$ \begin{equation} M=\frac{1}{T-1}\Bigg|X\Bigg(\frac{d}{df_t}>0\Bigg)-Y\Bigg(\frac{d}{df_t}<0\Bigg)\Bigg| \\ \end{equation} $$ </tex-math></div></p><p class=""><div class="disp-formula"><label>(2)</label><tex-math id="E2"> $$ \begin{equation} S=1-\frac{6\sum d_t^2}{T\left(T^2-1\right)} \\ \end{equation} $$ </tex-math></div></p><p class=""><div class="disp-formula"><label>(3)</label><tex-math id="E3"> $$ \begin{equation} R=\frac{1}{T}\sum\limits_{t=1}^{T}\exp\left(-\left|\frac{f_{t}-\tilde{f}_{t}}{f_{t}}\right|\right) \\ \end{equation} $$ </tex-math></div></p><p class=""><div class="disp-formula"><label>(4)</label><tex-math id="E4"> $$ \begin{equation} C=\frac{M+S+R}{3} \\ \end{equation} $$ </tex-math></div></p><p class="">respectively, where <inline-formula><tex-math id="M9">$$ T $$</tex-math></inline-formula> indicates the length of the degradation feature sequence, <inline-formula><tex-math id="M10">$$ f_{t} $$</tex-math></inline-formula> is the extracted value of the feature at time <inline-formula><tex-math id="M11">$$ t $$</tex-math></inline-formula>, <inline-formula><tex-math id="M12">$$ X(\cdot) $$</tex-math></inline-formula> denotes the number of positive derivatives in the degradation feature sequence, <inline-formula><tex-math id="M13">$$ Y(\cdot) $$</tex-math></inline-formula> denotes the number of negative derivatives in the degradation feature sequence, <inline-formula><tex-math id="M14">$$ \Sigma $$</tex-math></inline-formula> denotes the summation symbol, <inline-formula><tex-math id="M15">$$ d_{t} $$</tex-math></inline-formula> is the difference between the degradation feature index sequence and time series, <inline-formula><tex-math id="M16">$$ \exp(\cdot) $$</tex-math></inline-formula> represents an exponential function based on the natural constant <inline-formula><tex-math id="M17">$$ e $$</tex-math></inline-formula>, and <inline-formula><tex-math id="M18">$$ \tilde{f}_t $$</tex-math></inline-formula> is the value of the feature at time <inline-formula><tex-math id="M19">$$ t $$</tex-math></inline-formula> after sliding average.</p><p class="">Principal component analysis (PCA) was then used to reduce the dimensionality of the optimal feature subset, and the principal components were weighted according to their contribution degrees to generate a multidimensional principal component vector that retains the important information of the original optimal feature subset.</p><p class="">Finally, the MD<sup>[<a href="#b32" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b32">32</a>]</sup> was used to calculate the difference between the initial and subsequent samples in the generated multidimensional principal component vector. The obtained results were then smoothed using the exponential weighted moving average, yielding the CHI, which reflected the evolution of corrugation. The MD is calculated as follows:</p><p class=""><div class="disp-formula"><label>(5)</label><tex-math id="E5"> $$ \begin{equation} D_M\left(m,n\right)=\sqrt{\left(m-n\right)^T\text{K}^{-1}\left(m-n\right)} \\ \end{equation} $$ </tex-math></div></p><p class="">where <inline-formula><tex-math id="M20">$$ m $$</tex-math></inline-formula> and <inline-formula><tex-math id="M21">$$ n $$</tex-math></inline-formula> are the sample vectors; <inline-formula><tex-math id="M22">$$ K $$</tex-math></inline-formula> represents the covariance matrix of the corrugation evolution features.</p></div><div id="s2-2" class="article-Section"><h3 >2.2. Establishment of trend prediction model for rail corrugation</h3><p class="">To accurately predict the evolution trend of rail corrugation, we constructed a SA-BiTCN-BiGRU model. Using the initial corrugation data in the established CHI as the input, the subsequent CHI values were predicted.</p><p class="">The structure of the model is illustrated in <a href="#Figure2" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure2">Figure 2</a>. First, the bidirectional local features of the initial corrugation data were effectively extracted using a three-layer BiTCN to improve the receptive field and feature extraction capability of the model. Subsequently, based on the local features extracted by BiTCN, BiGRU was used for time-series prediction, and the output results were passed through the Leaky rectified linear unit (ReLU) nonlinear activation function and dropout regularization technology. The attention weight provided by the SA was then used to enhance the interpretability of the network. Finally, the multilayer perceptron (MLP) network output continuous prediction results and the error between them and the actual value was calculated to evaluate the prediction effect of the model.</p><div class="Figure-block" id="Figure2"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2024.20/image/Figure2" class="Article-img" alt="" target="_blank"><img alt="Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU" src="https://image.oaes.cc/80d245c1-6593-4d42-8e24-7a9b32dfa757/ir4020-2.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 2. Architecture of prediction model for rail corrugation trend.</p></div></div><div id="s2-2-1" class="article-Section"><h4 >2.2.1. BiTCN</h4><p class="">In this study, the constructed corrugation CHI is a continuous process that changes over time, and the data are closely related. To capture the features of the corrugation CHI over a wider range, a BiTCN was constructed to comprehensively consider the historical and forthcoming temporal information of the corrugation CHI. Additionally, multiple dilated causal convolution layers were stacked to improve the receptive field, effectively observe the change patterns in the rail corrugation data, and enhance the model's capacity to acquire key information. The structure of the BiTCN is shown in <a href="#Figure3" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure3">Figure 3</a>.</p><div class="Figure-block" id="Figure3"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2024.20/image/Figure3" class="Article-img" alt="" target="_blank"><img alt="Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU" src="https://image.oaes.cc/80d245c1-6593-4d42-8e24-7a9b32dfa757/ir4020-3.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 3. Schematic of BiTCN structure. BiTCN: Bidirectional temporal convolutional network.</p></div></div><p class="">As shown in <a href="#Figure3" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure3">Figure 3</a>, BiTCN consists of a forward and a reverse TCN residual block linked together. The model's output was the combined training result of the two blocks. Each residual block contained two layers of dilated causal convolution, which enlarged the receptive field of the network. The input sequence data were derived from the one-dimensional rail corrugation CHI, and the feature information at different scales was captured using the dilated convolution operation. A batch normalization layer was employed to stabilize the model training process. The Leaky ReLU activation function enabled the BiTCN module to train a deeper network while addressing dead neurons and vanishing gradient. Additionally, dropout regularization technology was added to reduce overfitting. To accommodate possible differences in the number of input and output channels in the model, a 1 × 1 convolution layer was added in each training direction for the residual connection, and the number of feature channels was adjusted to suit the feature representations of different levels.</p><p class="">The core concept of the dilated causal convolution involves the insertion of zero elements into the convolution kernel, which modifies the structure of the kernel and effectively expands the receptive field of the model. This enables each convolution output to encompass a broader range of time information, effectively mitigating vanishing gradient caused by numerous layers in the common convolution and enabling the model to extract more information on corrugation evolution<sup>[<a href="#b33" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b33">33</a>]</sup>. The internal structure of the dilated causal convolution is shown in <a href="#Figure4" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure4">Figure 4</a> and defined below:</p><p class=""><div class="disp-formula"><label>(6)</label><tex-math id="E6"> $$ \begin{equation} y=\sum\limits_{k=0}^{K-1}\omega\bigl[k\bigr]\cdot x\bigl[L-d\cdot k\bigr] \\ \end{equation} $$ </tex-math></div></p><div class="Figure-block" id="Figure4"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2024.20/image/Figure4" class="Article-img" alt="" target="_blank"><img alt="Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU" src="https://image.oaes.cc/80d245c1-6593-4d42-8e24-7a9b32dfa757/ir4020-4.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 4. Visualization of dilated causal convolution.</p></div></div><p class="">where <inline-formula><tex-math id="M23">$$ y $$</tex-math></inline-formula> is the output of the dilated causal convolution layer, <inline-formula><tex-math id="M24">$$ \omega[k] $$</tex-math></inline-formula> denotes the weight of the convolution kernel <inline-formula><tex-math id="M25">$$ k $$</tex-math></inline-formula>, <inline-formula><tex-math id="M26">$$ x[L-d\cdot k] $$</tex-math></inline-formula> denotes the value of the input sequence element, <inline-formula><tex-math id="M27">$$ L $$</tex-math></inline-formula> is the length of the input sequence, and <inline-formula><tex-math id="M28">$$ d $$</tex-math></inline-formula> represents the expansion rate.</p></div><div id="s2-2-2" class="article-Section"><h4 >2.2.2. BiGRU</h4><p class="">The GRU is a temporal prediction network proposed to alleviate the vanishing gradient problem of a recurrent neural network (RNN)<sup>[<a href="#b34" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b34">34</a>]</sup>. The evolution of corrugation is closely related to information from past and future data, and unidirectional GRU may fail to capture this bidirectional information transmission mode. Therefore, this study constructed a BiGRU to infer the relationship between past and future corrugation characteristics and the current corrugation amplitude to improve the model's sensitivity and predictive capability regarding dynamic changes in the time series of corrugation characteristics. The BiGRU is calculated using</p><p class=""><div class="disp-formula"><label>(7)</label><tex-math id="E7"> $$ \begin{equation} \begin{cases}\overrightarrow{h_n}=GRU\left(x_n,\overrightarrow{h}_{n-1}\right)\\\\\overleftarrow{h_n}=GRU\left(x_n,\overleftarrow{h}_{n-1}\right)\\\\h_n=\alpha_n\overrightarrow{h}_n+\beta_n\overleftarrow{h}_n+b_n&\end{cases} \\ \end{equation} $$ </tex-math></div></p><p class="">where <inline-formula><tex-math id="M29">$$ GRU(\cdot) $$</tex-math></inline-formula> denotes the gated cycle unit, <inline-formula><tex-math id="M30">$$ x_{n} $$</tex-math></inline-formula> is the input, <inline-formula><tex-math id="M31">$$ \overrightarrow{h_n} $$</tex-math></inline-formula> and <inline-formula><tex-math id="M32">$$ \overleftarrow{h_n} $$</tex-math></inline-formula> represent the output status of the forward and reverse hidden layers, respectively, <inline-formula><tex-math id="M33">$$ \alpha_{n} $$</tex-math></inline-formula> and <inline-formula><tex-math id="M34">$$ \beta_{n} $$</tex-math></inline-formula> are the corresponding output weights, and <inline-formula><tex-math id="M35">$$ b_{n} $$</tex-math></inline-formula> is the corresponding bias. The structure of the BiGRU is shown in <a href="#Figure5" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure5">Figure 5</a>.</p><div class="Figure-block" id="Figure5"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2024.20/image/Figure5" class="Article-img" alt="" target="_blank"><img alt="Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU" src="https://image.oaes.cc/80d245c1-6593-4d42-8e24-7a9b32dfa757/ir4020-5.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 5. Schematic of BiTCN structure. BiTCN: Bidirectional temporal convolutional network.</p></div></div><p class="">The entire network is composed of an input layer, two layers of GRUs in opposite directions, and an output layer. The input is the value after BiTCN feature extraction, and the output is determined based on the cycling training results of the BiGRU unit.</p></div><div id="s2-2-3" class="article-Section"><h4 >2.2.3. SA mechanism</h4><p class="">As a variant of the attention mechanism, SA<sup>[<a href="#b35" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b35">35</a>]</sup> is mainly used to process serial data such as the rail corrugation time-series data used in this study. The network can calculate the attention weights of various positions at different time steps to improve its ability to obtain key information and integrate the content of all time steps. This study introduced and applied the SA mechanism to the process of model trend prediction, which was designed to improve the model's dependence on different locations in the input ripple CHI sequence. This allowed the model to better understand the internal correlations between the corrugation data at each moment, significantly improving its predictive performance. This technology can provide reliable decision-making support for the maintenance and management of railway systems. <a href="#Figure6" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure6">Figure 6</a> shows the structure of the SA.</p><div class="Figure-block" id="Figure6"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2024.20/image/Figure6" class="Article-img" alt="" target="_blank"><img alt="Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU" src="https://image.oaes.cc/80d245c1-6593-4d42-8e24-7a9b32dfa757/ir4020-6.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 6. Illustration of SA mechanism. SA: Self-attention.</p></div></div><p class="">As depicted in <a href="#Figure6" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure6">Figure 6</a>, the structure initially computes and packages the query, key, and value vectors of all input matrices as matrices. The query and key vectors were used to perform a nonlinear transformation. The dot product and masking operations standardized the query and key vectors, masked invalid information, and generated an attention score. The mapping matrix of the attention score was then obtained after normalization using the softmax operation and multiplied by the value vector after identity mapping to acquire the weight output.</p></div></div><div id="s2-3" class="article-Section"><h3 >2.3. Model hyperparameter optimization based on MICPO algorithm</h3><p class="">Certain hyperparameters significantly affected the predictive performance of the proposed model. For example, the convolution kernel size determined the capability of the model to capture corrugation characteristics in the time dimension. The number of BiGRU hidden layer units determines the complexity and learning ability of the network. To prevent the adverse effects of manual intervention in the selection of model hyperparameters, optimization algorithms are necessary to adaptively identify the most suitable model hyperparameters.</p><p class="">Consequently, model hyperparameter optimization was performed using the crested porcupine optimizer (CPO)<sup>[<a href="#b36" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b36">36</a>]</sup> algorithm. This algorithm simulates four different defense strategies when a crested porcupine (CP) engages in defense against predators. The first two strategies, sight and sound, represent the exploration phase of the algorithm; the last two strategies, odor and physical-attack, represent the exploitation phase of the algorithm. Different defense strategies have distinct optimization effects on various hyperparameters, guiding the algorithm to identify the optimal hyperparameters for the model. However, the original algorithm has certain limitations, such as decreasing population diversity and the tendency to get trapped in local optimality in the later stages of a search, leading to an inaccurate selection of hyperparameters. Therefore, a multi-strategy improvement method was constructed to optimize the initialization mode and defense strategy of the CPO algorithm to acquire better model hyperparameters and enhance the prediction accuracy of the model on the evolution trend of rail corrugation. The detailed improvement strategies for the CPO algorithm are discussed in the following subsections.</p><div id="s2-3-1" class="article-Section"><h4 >2.3.1. Improved tent chaos map</h4><p class="">In the algorithm initialization stage, an improved tent map was employed to generate chaotic sequences and address issues related to the reduction of the CP population and its tendency to converge into the local optimal solution when the CPO algorithm approached the global optimum<sup>[<a href="#b37" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b37">37</a>]</sup>. This method introduced random variables into a traditional tent-chaos map. Thus, the diversity of the CP individuals was increased, and the chaotic sequence was prevented from falling into unstable periodic points during the iterative process defined as follows:</p><p class=""><div class="disp-formula"><label>(8)</label><tex-math id="E8"> $$ \begin{equation} X_{i,j+1}=\begin{cases}\frac{X_{i,j}}{tent}+rand\left(0,1\right),&\quad0\leq{X}_{i,j}\leq tent\\\\\frac{1-X_{i,j}}{1-tent}+rand\left(0,1\right),&\quad{tent}<{X}_{i,j}\leq1\end{cases} \\ \end{equation} $$ </tex-math></div></p><p class="">where <inline-formula><tex-math id="M36">$$ i $$</tex-math></inline-formula> and <inline-formula><tex-math id="M37">$$ j $$</tex-math></inline-formula> represent the CP population number and current dimension, respectively; <inline-formula><tex-math id="M38">$$ tent $$</tex-math></inline-formula> denotes the chaos coefficient, and <inline-formula><tex-math id="M39">$$ rand\begin{pmatrix}0,1\end{pmatrix} $$</tex-math></inline-formula> represents a random number between 0 and 1.</p></div><div id="s2-3-2" class="article-Section"><h4 >2.3.2. Golden sine strategy</h4><p class="">In this study, the golden sine strategy<sup>[<a href="#b38" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b38">38</a>]</sup> was incorporated into the CPO algorithm to enlarge its search space and address the lack of information exchange between CP individuals in the original algorithm, thereby improving the algorithm's ability for global optimization defined as follows:</p><p class=""><div class="disp-formula"><label>(9)</label><tex-math id="E9"> $$ \begin{equation} X_{i,j}^{t+1}=X_{i,j}^{t}\times\left|\sin\left(D_{1}\right)\right|+D_{2}\times\sin\left(D_{1}\right)\times\left|x_{1}\times X_{i,j}^{t}-x_{2}\times X_{i,j}^{t}\right| \\ \end{equation} $$ </tex-math></div></p><p class="">where <inline-formula><tex-math id="M40">$$ t $$</tex-math></inline-formula> denotes the number of iterations. After using this formula to improve the position update strategy of the algorithm, all CP individuals exchanged information with the optimal individuals in each exploration phase. Simultaneously, the golden section coefficient gradually reduced the search space of the CP individuals. By controlling the moving distance and direction of the CP individuals, the CPO algorithm was optimized, further coordinating the algorithm's global exploration and local exploitation abilities.</p></div><div id="s2-3-3" class="article-Section"><h4 >2.3.3. Adaptive weight strategy</h4><p class="">When executing the third defense strategy, the search step of the CP individual was not set in the original algorithm, resulting in excessive freedom while running the algorithm. The adaptive weight strategy can dynamically adjust the optimal position<sup>[<a href="#b39" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b39">39</a>]</sup>, thereby effectively enhancing the convergence effect and local exploitation ability of the CPO algorithm. This adjustment ensures that individuals with CP maintain a relatively safe distance from predators while executing the third defense strategy. Therefore, this study constructed an adaptive strategy that adjusted the weight coefficient <inline-formula><tex-math id="M41">$$ \omega $$</tex-math></inline-formula> based on the iteration count, allowing CP individuals to utilize different weights for optimal search lengths at different stages. The <inline-formula><tex-math id="M42">$$ \omega $$</tex-math></inline-formula> is obtained as</p><p class=""><div class="disp-formula"><label>(10)</label><tex-math id="E10"> $$ \begin{equation} \omega=1-\cosh\left(\left(\exp(t/T_{\max})\right)/\exp(1)-1\right)^2 \\ \end{equation} $$ </tex-math></div></p><p class="">where <inline-formula><tex-math id="M43">$$ \text{cosh()} $$</tex-math></inline-formula> denotes the hyperbolic cosine function, and <inline-formula><tex-math id="M44">$$ T_{\max} $$</tex-math></inline-formula> denotes the maximum number of iterations.</p></div><div id="s2-3-4" class="article-Section"><h4 >2.3.4. Variable spiral search strategy</h4><p class="">Inspired by the whale optimization algorithm (WOA)<sup>[<a href="#b40" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b40">40</a>]</sup>, the variable spiral search strategy adjusts the original spiral parameters to become variable parameters that change with each iteration. This adjustment allows the algorithm to perform extensive searches in the early phase and an elaborate exploration of a small area in the late stage<sup>[<a href="#b41" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b41">41</a>]</sup>, enhancing its local exploitation ability in the fourth defense strategy. In this study, by constructing a variable spiral search strategy, CP individuals continued to search nearby after reaching the local optimal solution. This approach compensates for the unclear convergence effect of the original CPO during local exploration, which prevents deviations in the prediction accuracy of the model in the late stages of rail corrugation development. This strategy is established as</p><p class=""><div class="disp-formula"><label>(11)</label><tex-math id="E11"> $$ \begin{equation} Z=X_{best}\begin{pmatrix}t\end{pmatrix}\times\begin{pmatrix}\exp\begin{pmatrix}zl\end{pmatrix}\times\cos\begin{pmatrix}2\pi l\end{pmatrix}\end{pmatrix}+X_{best}\begin{pmatrix}t\end{pmatrix} \\ \end{equation} $$ </tex-math></div></p><p class=""><div class="disp-formula"><label>(12)</label><tex-math id="E12"> $$ \begin{equation} z=\exp\bigl(k\cos\bigl(\pi t/T_{\max}\bigr)\bigr) \\ \end{equation} $$ </tex-math></div></p><p class="">where <inline-formula><tex-math id="M45">$$ X_{best} $$</tex-math></inline-formula> denotes the best fitness value, <inline-formula><tex-math id="M46">$$ l $$</tex-math></inline-formula> represents a random number between -1 and 1, and <inline-formula><tex-math id="M47">$$ k $$</tex-math></inline-formula> represents a variable parameter that should be set according to the specific strategy.</p><p class="">Based on the above analysis, a flowchart of the MICPO algorithm is constructed [<a href="#Figure7" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure7">Figure 7</a>], where <inline-formula><tex-math id="M48">$$ N $$</tex-math></inline-formula> and <inline-formula><tex-math id="M49">$$ T_{max} $$</tex-math></inline-formula> represent the population size and maximum number of function evaluations, respectively. <inline-formula><tex-math id="M50">$$ T_{f} $$</tex-math></inline-formula> indicates a constant between 0 and 1, <inline-formula><tex-math id="M51">$$ t $$</tex-math></inline-formula> denotes the number of current iterations, and <inline-formula><tex-math id="M52">$$ i $$</tex-math></inline-formula> is the current <inline-formula><tex-math id="M53">$$ i $$</tex-math></inline-formula><inline-formula><tex-math id="M54">$$ th $$</tex-math></inline-formula> individual. In the first iteration, all CP individuals passed through the position of the initialization solution and adopted a defense strategy to obtain the current optimal candidate solution. Subsequently, the algorithm entered the next iteration. First, the defense factor and the population number <inline-formula><tex-math id="M55">$$ N $$</tex-math></inline-formula> were updated. Then, the CP individuals continue to search for the best candidate solution of the model according to the selected defense strategy. This process was repeated until the iterations were complete. Consequently, the optimal solution, which represents the best parameter of the model, was obtained and substituted into the SA-BiTCN-BiGRU hybrid network to optimize the prediction performance of the model.</p><div class="Figure-block" id="Figure7"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2024.20/image/Figure7" class="Article-img" alt="" target="_blank"><img alt="Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU" src="https://image.oaes.cc/80d245c1-6593-4d42-8e24-7a9b32dfa757/ir4020-7.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 7. Flowchart of MICPO algorithm. MICPO: Multi-strategy improved crested porcupine optimizer.</p></div></div></div></div><div id="s2-4" class="article-Section"><h3 >2.4. Algorithm validation</h3><p class="">In this study, six benchmark functions were used to conduct the optimization experiments. The MICPO algorithm was compared with the CPO<sup>[<a href="#b36" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b36">36</a>]</sup>, WOA<sup>[<a href="#b40" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b40">40</a>]</sup>, rime optimization algorithm (RIME)<sup>[<a href="#b42" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b42">42</a>]</sup>, grey wolf optimizer (GWO)<sup>[<a href="#b43" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b43">43</a>]</sup>, and dung-beetle optimizer (DBO)<sup>[<a href="#b44" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b44">44</a>]</sup> algorithm to observe their optimal fitness values and convergence speed within a specified number of iterations, and verify the improvement effect of MICPO on the original CPO. <a href="#Table1" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Table1">Table 1</a> provides a detailed definition of the benchmark functions. F1-F3 are single-peak functions used to evaluate the local search capability of the algorithm. F4 is a multipeak function with multiple local optimal values and requires a higher convergence performance of the algorithm. This function has important reference significance in the evaluation algorithm. F5 and F6 are the combined benchmark functions used to evaluate the global exploitation capacity of an algorithm. In this study, the population size of the experimental algorithm was set to 30, and each algorithm was optimized 100 times.</p><div id="Table1" class="Figure-block"><div class="table-note"><span class="">Table 1</span><p class="">Detailed information on benchmark function</p></div><div class="table-responsive article-table"><table class="a-table"><thead><tr><td style="class:table_top_border" align="center"><b>ID</b></td><td style="class:table_top_border" align="center"><b>Benchmark function</b></td><td style="class:table_top_border" align="center"><b>Domain and dimensions</b></td><td style="class:table_top_border" align="center"><b>Optimal value</b></td></tr></thead><tbody><tr><td style="class:table_top_border2" align="center">F1</td><td style="class:table_top_border2" align="center"><inline-formula><tex-math id="M56">$$ f_1(x)=\sum\limits_{i=1}^{n}{x_{i}^{2}} $$</tex-math></inline-formula></td><td style="class:table_top_border2" align="center"><inline-formula><tex-math id="M57">$$ \begin{bmatrix}-100, 100\end{bmatrix}^{30} $$</tex-math></inline-formula></td><td style="class:table_top_border2" align="center">0</td></tr><tr><td align="center">F2</td><td align="center"><inline-formula><tex-math id="M58">$$ f_1(x) = \sum\limits_{i = 1}^n {x_i^2} $$</tex-math></inline-formula></td><td align="center"><inline-formula><tex-math id="M59">$$ \begin{bmatrix}-100, 100\end{bmatrix}^{30} $$</tex-math></inline-formula></td><td align="center">0</td></tr><tr><td align="center">F3</td><td align="center"><inline-formula><tex-math id="M60">$$ f_3(x) = \sum\limits_{i = 1}^n {ix_i^4 + \text{random}[0,1)} $$</tex-math></inline-formula></td><td align="center"><inline-formula><tex-math id="M61">$$ \begin{bmatrix}-1.28, 1.28\end{bmatrix}^{30} $$</tex-math></inline-formula></td><td align="center">0</td></tr><tr><td align="center">F4</td><td align="center"><inline-formula><tex-math id="M62">$$ f_4(x) = - 20\exp \left( { - 0.2\sqrt {\frac{1}{n}\sum\limits_{i = 1}^n {x_i^2} } } \right) - \exp \left( {\frac{1}{n}\sum\limits_{i = 1}^n {\cos } (2\pi {x_i})} \right) + 20 + e $$</tex-math></inline-formula></td><td align="center"><inline-formula><tex-math id="M63">$$ \begin{bmatrix}-32, 32\end{bmatrix}^{30} $$</tex-math></inline-formula></td><td align="center">0</td></tr><tr><td align="center">F5</td><td align="center"><inline-formula><tex-math id="M64">$$ f_5(x) = \sum\limits_{i = 1}^n {\left[ {{a_i} - \frac{{{x_1}(b_i^2 + {b_1}{x_2})}}{{b_i^2 + {b_1}{x_3} + {x_4}}}} \right]} $$</tex-math></inline-formula></td><td align="center"><inline-formula><tex-math id="M65">$$ \begin{bmatrix}-5, 5\end{bmatrix}^{4} $$</tex-math></inline-formula></td><td align="center"><inline-formula><tex-math id="M66">$$ 3.075\times10^{-4} $$</tex-math></inline-formula></td></tr><tr><td style="class:table_bottom_border" align="center">F6</td><td style="class:table_bottom_border" align="center"><inline-formula><tex-math id="M67">$$ f_6(x) = - \sum\limits_{i = 1}^{10} {{{\left[ {(x - {a_i}){{(x - {a_i})}^T} + {c_i}} \right]}^{ - 1}}} $$</tex-math></inline-formula></td><td style="class:table_bottom_border" align="center"><inline-formula><tex-math id="M68">$$ \begin{bmatrix}0, 10\end{bmatrix}^{4} $$</tex-math></inline-formula></td><td style="class:table_bottom_border" align="center">-10</td></tr></tbody></table></div><div class="table_footer"></div></div><p class="">As shown in <a href="#Figure8" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure8">Figure 8</a>, the convergence performance of the MICPO algorithm is effectively proven.</p><div class="Figure-block" id="Figure8"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2024.20/image/Figure8" class="Article-img" alt="" target="_blank"><img alt="Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU" src="https://image.oaes.cc/80d245c1-6593-4d42-8e24-7a9b32dfa757/ir4020-8.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 8. Convergence curves of different algorithms under different benchmark functions. (A-F) correspond to benchmark functions F1-F6, respectively.</p></div></div><p class="">From the convergence curve presented in <a href="#Figure8" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure8">Figure 8</a>, the CPO algorithm exhibits poor convergence performance and easily falls into the local optima, indicating that improvements in the CPO algorithm are necessary. In the unimodal function test shown in <a href="#Figure8" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure8">Figure 8A</a>-<a href="#Figure8" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure8">C</a>, the RIME, WOA, and the other optimization algorithms fell into local optima and slowly converged, indicating that the MICPO algorithm has certain competitive advantages over other optimization algorithms in solving unimodal high-dimensional functions. In the multipeak test function F4, the MICPO algorithm [<a href="#Figure8" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure8">Figure 8D</a>], demonstrates an advantage by being the closest to the optimal solution within the specified number of iterations, which validates its effectiveness in improving the CPO algorithm, as well as its superiority in search accuracy and convergence speed. In the combined function test shown in <a href="#Figure8" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure8">Figure 8E</a> and <a href="#Figure8" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure8">F</a>, the MICPO and CPO algorithms demonstrate superior convergence performance compared to the RIME algorithm and other optimization algorithms, indicating their advancement in global optimization.</p></div></div><div id="s3" class="article-Section"><h2 >3. RESULTS</h2><div id="s3-1" class="article-Section"><h3 >3.1. Experimental setup and rail corrugation dataset preprocessing</h3><p class="">First, the code was written and debugged on a PyCharm platform, and the running environment consisted of a processor (Intel i7-12700H), 16 GB of random-access memory (RAM), a graphics card (RTX 3060), and a software environment with TensorFlow 2.13.0 and Python 3.9.18. The experimental data in this study were actual measurement data from a railway section in China. A track inspection car was used to collect vibration signals from a typical steel rail segment with corrugation, covering damage from slight to severe stages. These signals demonstrate the progression of corrugation damage<sup>[<a href="#b21" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b21">21</a>]</sup>. We collected 98 vibration samples on-site at the same time interval throughout the entire lifecycle of the rail after several months of continuous periodic testing, with each vibration sample containing 3, 000 sample points; therefore, the original sample contained 98 × 3, 000 data points. These data points represent the initial corrugation on the rail surface to the rail scrap. The overall vibration amplitude gradually increased with collection times, indicating that the deterioration degree of corrugation damage was worsening, reflecting the evolution of rail corrugation damage from budding to deterioration.</p><p class="">First, each collected sample was subjected to multidomain feature extraction to obtain 26 feature indicators that reflected the evolution of corrugation. The dimensions of the samples were 98 × 26. Subsequently, the <inline-formula><tex-math id="M69">$$ M $$</tex-math></inline-formula>, <inline-formula><tex-math id="M70">$$ S $$</tex-math></inline-formula>, and <inline-formula><tex-math id="M71">$$ R $$</tex-math></inline-formula> of each feature index were calculated; <inline-formula><tex-math id="M72">$$ C $$</tex-math></inline-formula> was used to adaptively screen out the eight features with higher scores, and the corrugation optimal feature subset with a sample dimension of 98 × 8 was obtained. PCA was used to fuse the optimal feature subset, resulting in a two-dimensional principal component vector with a total contribution rate of 97%. The sample dimensions of the corrugation data were 98 × 2. Finally, the MD was used to calculate the difference between the first column of the sample data and the subsequent 97 columns of sample data, resulting in 98 × 1 one-dimensional data. The corrugation CHI was obtained after smoothing to minimize the negative impact of outliers on the prediction of the evolution trend of rail corrugation.</p></div><div id="s3-2" class="article-Section"><h3 >3.2. Validation of the CHI construction method</h3><p class="">To demonstrate the effectiveness and advantages of the method proposed for constructing the corrugation CHI, several commonly used methods for constructing health indicators were selected for comparison, including the RMS, PCA, and locally linear embedding (LLE) fusion indicators. Two fusion indicators were constructed using the optimal feature subset described in Section 3.1. The rail corrugation health indicator constructed using these four methods after smoothing is shown in <a href="#Figure9" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure9">Figure 9</a>.</p><div class="Figure-block" id="Figure9"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2024.20/image/Figure9" class="Article-img" alt="" target="_blank"><img alt="Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU" src="https://image.oaes.cc/80d245c1-6593-4d42-8e24-7a9b32dfa757/ir4020-9.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 9. Health indicators of rail corrugation constructed by different methods. (A-D) correspond to RMS, PCA, LLE and CHI methods, respectively. RMS: Root mean square; PCA: principal component analysis; LLE: locally linear embedding; CHI: comprehensive health indicator.</p></div></div><p class=""><a href="#Figure9" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure9">Figure 9</a> shows that these indicators are relatively sensitive to changes in the initial corrugation damage. However, the RMS indicator exhibits a larger overall fluctuation range, with the index value declining in the later stages of the corrugation evolution and deviating from the actual situation. The amplitude of the PCA indicator fluctuates significantly between the middle and late stages. The LLE indicator oscillates excessively in the early stages and becomes more stable in the middle and late stages, which is different from the actual situation. However, the CHI constructed in this study showed a better overall trend with fewer fluctuations. The indicator shows a sudden increase when the corrugation damage approached a qualitative change in the later stage, which aligns with the actual evolution law of on-site rail corrugation damage. The corrugation health indicator constructed by CHI is more consistent with the changing trend of the real-world data on rail corrugation vibration signals.</p><p class="">Furthermore, the <inline-formula><tex-math id="M73">$$ M $$</tex-math></inline-formula>, <inline-formula><tex-math id="M74">$$ S $$</tex-math></inline-formula>, <inline-formula><tex-math id="M75">$$ R $$</tex-math></inline-formula>, and <inline-formula><tex-math id="M76">$$ C $$</tex-math></inline-formula> [established by Equations (1)-(4)] were used to evaluate the HIs constructed using the four different methods. The results are listed in <a href="#Table2" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Table2">Table 2</a>.</p><div id="Table2" class="Figure-block"><div class="table-note"><span class="">Table 2</span><p class="">Evaluation results of health indicators</p></div><div class="table-responsive article-table"><table class="a-table"><thead><tr><td align="center" style="class:table_top_border"><b>Indicator</b></td><td align="center" style="class:table_top_border"><b><i>M</i></b></td><td align="center" style="class:table_top_border"><b><i>S</i></b></td><td align="center" style="class:table_top_border"><b><i>R</i></b></td><td align="center" style="class:table_top_border"><b><i>C</i></b></td></tr></thead><tbody><tr><td style="class:table_top_border2" align="center">RMS</td><td style="class:table_top_border2" align="center">0.1134</td><td style="class:table_top_border2" align="center">0.8336</td><td style="class:table_top_border2" align="center">0.9043</td><td style="class:table_top_border2" align="center">0.6171</td></tr><tr><td align="center">PCA</td><td align="center">0.1546</td><td align="center">0.8706</td><td align="center">0.9021</td><td align="center">0.6424</td></tr><tr><td align="center">LLE</td><td align="center">0.1753</td><td align="center">0.8983</td><td align="center">0.9098</td><td align="center">0.6611</td></tr><tr><td style="class:table_bottom_border" align="center">Proposed indicator</td><td style="class:table_bottom_border" align="center">0.2165</td><td style="class:table_bottom_border" align="center">0.9322</td><td style="class:table_bottom_border" align="center">0.928</td><td style="class:table_bottom_border" align="center">0.6922</td></tr></tbody></table></div><div class="table_footer"><div class="table_footer_note"><p class="para">RMS: Root mean square; PCA: principal component analysis; LLE: linear embedding.</p></div></div></div><p class="">Through the comparison of various indicators in <a href="#Table2" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Table2">Table 2</a>, the constructed CHI achieved optimal performance in all cases, with the highest comprehensive evaluation function <inline-formula><tex-math id="M77">$$ C $$</tex-math></inline-formula>. Therefore, this indicator is considered suitable for reflecting the evolution trend of rail corrugation.</p></div><div id="s3-3" class="article-Section"><h3 >3.3. Performance evaluation indicators</h3><p class="">The root mean square error (RMSE), mean square error (MSE), and mean absolute error (MAE) were used to evaluate the performance of the model. These indices reflect the prediction effect by calculating the error between the predicted and true CHI values. Simultaneously, to address inconsistencies among the different indicator dimensions, <inline-formula><tex-math id="M78">$$ R^{2} $$</tex-math></inline-formula> was added as an evaluation criterion. The indices are estimated using</p><p class=""><div class="disp-formula"><label>(13)</label><tex-math id="E13"> $$ \begin{equation} {RMSE = \sqrt {\frac{1}{N}\mathop \sum \nolimits_{i = 1}^N {{({x_i} - {y_i})}^2}} } \\ \end{equation} $$ </tex-math></div></p><p class=""><div class="disp-formula"><label>(14)</label><tex-math id="E14"> $$ \begin{equation} MSE=\frac{\sum_{\mathrm{i}=1}^{\mathrm{N}}\left(x_i-y_i\right)^2}N \\ \end{equation} $$ </tex-math></div></p><p class=""><div class="disp-formula"><label>(15)</label><tex-math id="E15"> $$ \begin{equation} MAE=\frac{\sum_{i=1}^N\left|x_i-y_i\right|}{N} \\ \end{equation} $$ </tex-math></div></p><p class=""><div class="disp-formula"><label>(16)</label><tex-math id="E16"> $$ \begin{equation} R^{2}=1-\frac{\sum_{i=1}^{N}(x_{i}-y_{i})^{2}}{\sum_{i=1}^{N}(x_{i}-\overline{x}_{i})^{2}} \\ \end{equation} $$ </tex-math></div></p><p class="">where <inline-formula><tex-math id="M79">$$ {x_i} $$</tex-math></inline-formula> represents the true corrugation CHI value, <inline-formula><tex-math id="M80">$$ {y_i} $$</tex-math></inline-formula> represents the predicted value by the model, <inline-formula><tex-math id="M81">$$ {N} $$</tex-math></inline-formula> is the data length of the corrugation CHI, and <inline-formula><tex-math id="M82">$$ \bar{x}_i $$</tex-math></inline-formula> denotes the average of the true value.</p></div><div id="s3-4" class="article-Section"><h3 >3.4. Predictive experimental analysis of rail corrugation</h3><p class="">After obtaining the CHI of the corrugation damage according to Section 3.1, the corrugation CHI data with a length of 98 can be expressed as <inline-formula><tex-math id="M83">$$ \{x_1,x_2,\cdots,x_{98}\} $$</tex-math></inline-formula>. Our study used 75% of the data as the training set and the remainder as the test set; thus, the training set was <inline-formula><tex-math id="M84">$$ \{x_1,x_2,\cdots,x_{n}\} $$</tex-math></inline-formula> and the test set was <inline-formula><tex-math id="M85">$$ \{x_{n+1},x_{n+2},\cdots,x_{98}\} $$</tex-math></inline-formula>. The SA-BiTCN-BiGRU model was trained using the training set, whereas the test set was used to verify the effect of the model in predicting the evolution trend of rail corrugation. Subsequently, according to the input step length <inline-formula><tex-math id="M86">$$ t $$</tex-math></inline-formula> of the prediction model, the single-step sliding window approach was used for forecasting, using the corrugation initial data from the CHI to predict the subsequent evolution of the corrugation. For example, the input of the first sample was <inline-formula><tex-math id="M87">$$ \begin{Bmatrix}x_1,x_2,\cdots,x_t\end{Bmatrix} $$</tex-math></inline-formula>, yielding the prediction result of <inline-formula><tex-math id="M88">$$ y_{t+1} $$</tex-math></inline-formula>. Then, the prediction was gradually conducted to obtain the prediction result <inline-formula><tex-math id="M89">$$ y_{98} $$</tex-math></inline-formula> of the last sample. The error between the predicted value of the model and the input CHI was calculated to evaluate the prediction performance of the model. Other variables that may have affected the experimental results were controlled to ensure that the observed changes were caused by the proposed method. <a href="#Table3" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Table3">Table 3</a> presents the network model parameters used for predicting the evolution trend of the rail corrugation.</p><div id="Table3" class="Figure-block"><div class="table-note"><span class="">Table 3</span><p class="">Main parameter settings of proposed network</p></div><div class="table-responsive article-table"><table class="a-table"><thead><tr><td style="class:table_top_border" align="center"><b>Parameter</b></td><td style="class:table_top_border" align="center"><b>Value</b></td></tr></thead><tbody><tr><td style="class:table_top_border2" align="center">Epochs</td><td style="class:table_top_border2" align="center">1000</td></tr><tr><td align="center">Batch_size</td><td align="center">128</td></tr><tr><td align="center">Optimizer</td><td align="center">Adam</td></tr><tr><td align="center">Leaky rate</td><td align="center">0.01</td></tr><tr><td align="center">Learning rate</td><td align="center"><inline-formula><tex-math id="M90">$$ [1\times10^{-4},1\times10^{-2}] $$</tex-math></inline-formula></td></tr><tr><td align="center">Dropout rate</td><td align="center"><inline-formula><tex-math id="M91">$$ [1\times10^{-3},1\times10^{-2}] $$</tex-math></inline-formula></td></tr><tr><td align="center">Kernel size</td><td align="center">[2, 7]</td></tr><tr><td align="center">Number of filters</td><td align="center">[8, 128]</td></tr><tr><td style="class:table_bottom_border" align="center">Number of BiGRU hidden unit</td><td style="class:table_bottom_border" align="center">[8, 128]</td></tr></tbody></table></div><div class="table_footer"><div class="table_footer_note"><p class="para">BiGRU: Bidirectional gated recurrent unit.</p></div></div></div><div id="s3-4-1" class="article-Section"><h4 >3.4.1. Ablation experiment</h4><p class="">A comprehensive quantitative analysis of the structure and function of the proposed network was conducted to highlight the effects of each module on the MICPO-SA-BiTCN-BiGRU network. The results are summarized in <a href="#Table4" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Table4">Table 4</a>.</p><div id="Table4" class="Figure-block"><div class="table-note"><span class="">Table 4</span><p class="">Ablation experiment prediction errors</p></div><div class="table-responsive article-table"><table class="a-table"><thead><tr><td style="class:table_top_border" align="center"><b>Prediction model</b></td><td style="class:table_top_border" align="center"><b>RMSE</b></td><td style="class:table_top_border" align="center"><b>MSE</b></td><td style="class:table_top_border" align="center"><b>MAE</b></td><td style="class:table_top_border" align="center"><b><i>R</i><sup>2</sup></b></td></tr></thead><tbody><tr><td style="class:table_top_border2" align="center">TCN</td><td style="class:table_top_border2" align="center">0.415</td><td style="class:table_top_border2" align="center">0.172</td><td style="class:table_top_border2" align="center">0.309</td><td style="class:table_top_border2" align="center">0.82</td></tr><tr><td align="center">TCN-GRU</td><td align="center">0.341</td><td align="center">0.117</td><td align="center">0.227</td><td align="center">0.878</td></tr><tr><td align="center">BiTCN-BiGRU</td><td align="center">0.315</td><td align="center">0.099</td><td align="center">0.194</td><td align="center">0.896</td></tr><tr><td align="center">SA-BiTCN-BiGRU</td><td align="center">0.239</td><td align="center">0.057</td><td align="center">0.13</td><td align="center">0.94</td></tr><tr><td align="center">CPO-SA-BiTCN-BiGRU</td><td align="center">0.171</td><td align="center">0.029</td><td align="center">0.119</td><td align="center">0.969</td></tr><tr><td style="class:table_bottom_border" align="center">Proposed model</td><td style="class:table_bottom_border" align="center">0.119</td><td style="class:table_bottom_border" align="center">0.014</td><td style="class:table_bottom_border" align="center">0.095</td><td style="class:table_bottom_border" align="center">0.985</td></tr></tbody></table></div><div class="table_footer"><div class="table_footer_note"><p class="para">RMSE: Root mean square error; MSE: mean square error; MAE: mean absolute error; TCN: temporal convolutional network; GRU: gated recurrent unit; BiTCN: bidirectional temporal convolutional network; BiGRU: bidirectional gated recurrent unit; SA: self-attention; CPO: crested porcupine optimizer.</p></div></div></div><p class="">From <a href="#Table4" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Table4">Table 4</a>, the RMSE, MSE, and MAE decreased by 17.8%, 32%, and 26.5%, respectively, from TCN to TCN-GRU, whereas <inline-formula><tex-math id="M92">$$ R^{2} $$</tex-math></inline-formula> increased by 7.1%. If a bidirectional network structure (BiTCN-BiGRU model) was added, the RMSE, MSE and MAE further decreased by 7.6%, 15.4%, and 14.5%, respectively, whereas <inline-formula><tex-math id="M93">$$ R^{2} $$</tex-math></inline-formula> increased by 2.1%. This indicates that the structure further improved its prediction by considering the information on the forward and backward evolution of rail corrugation. When SA was introduced into the BiTCN-BiGRU model, the RMSE, MSE, and MAE decreased by 24.1%, 42.4%, and 33%, respectively, and <inline-formula><tex-math id="M94">$$ R^{2} $$</tex-math></inline-formula> increased by 4.91%. This indicates that the introduction of SA improved the feature expression capability of the network and reduced its dependence on irrelevant information. To reduce the impact of the artificial selection of network hyperparameters on the prediction results, the CPO algorithm was added for model optimization. Consequently, the RMSE, MSE, and MAE decreased by 28.5%, 49.1%, and 8.5%, respectively, and <inline-formula><tex-math id="M95">$$ R^{2} $$</tex-math></inline-formula> increased by 3.1%. Subsequently, the search strategy of the original CPO algorithm was improved to effectively alleviate the problem of local convergence. Consequently, the RMSE, MSE, and MAE decreased by 30.4%, 51.7%, and 20.2%, respectively, and <inline-formula><tex-math id="M96">$$ R^{2} $$</tex-math></inline-formula> increased by 1.7%, indicating that the optimization algorithm and its improved strategy were effective for model prediction.</p><p class="">The visualization results of the model ablation experiment presented in <a href="#Figure10" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure10">Figure 10</a> show that the proposed method (brown line) closely matches the true value (green line), particularly during the model testing phase, thus effectively predicting the evolution process of corrugation in the later stages of development. Additionally, the proposed method shows a higher local prediction accuracy compared to the other models.</p><div class="Figure-block" id="Figure10"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2024.20/image/Figure10" class="Article-img" alt="" target="_blank"><img alt="Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU" src="https://image.oaes.cc/80d245c1-6593-4d42-8e24-7a9b32dfa757/ir4020-10.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 10. Model ablation experiment: prediction results of evolution trend of rail corrugation.</p></div></div></div><div id="s3-4-2" class="article-Section"><h4 >3.4.2. Comparison experiment</h4><p class="">To verify the timeliness of the MICPO-SA-BiTCN-BiGRU network model in predicting the evolution trend of rail corrugation, we used the network models from recently published studies to predict the evolution trend of rail corrugation and quantitatively analyze and compare the predicted results with those of the proposed model. The results are listed in <a href="#Table5" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Table5">Table 5</a>.</p><div id="Table5" class="Figure-block"><div class="table-note"><span class="">Table 5</span><p class="">Comparison experiment prediction errors</p></div><div class="table-responsive article-table"><table class="a-table"><thead><tr><td style="class:table_top_border" align="center"><b>Prediction model</b></td><td style="class:table_top_border" align="center"><b>RMSE</b></td><td style="class:table_top_border" align="center"><b>MSE</b></td><td style="class:table_top_border" align="center"><b>MAE</b></td><td style="class:table_top_border" align="center"><b><i>R</i><sup>2</sup></b></td></tr></thead><tbody><tr><td style="class:table_top_border2" align="center">TCN-GRU-attention<sup>[<a href="#b20" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b20">20</a>]</sup></td><td style="class:table_top_border2" align="center">0.351</td><td style="class:table_top_border2" align="center">0.123</td><td style="class:table_top_border2" align="center">0.223</td><td style="class:table_top_border2" align="center">0.872</td></tr><tr><td align="center">CNN-BiGRU-attention<sup>[<a href="#b23" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b23">23</a>]</sup></td><td align="center">0.36</td><td align="center">0.13</td><td align="center">0.276</td><td align="center">0.864</td></tr><tr><td align="center">CNN-GRU<sup>[<a href="#b45" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b45">45</a>]</sup></td><td align="center">0.448</td><td align="center">0.201</td><td align="center">0.351</td><td align="center">0.79</td></tr><tr><td align="center">CNN-LSTM-attention<sup>[<a href="#b46" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b46">46</a>]</sup></td><td align="center">0.377</td><td align="center">0.142</td><td align="center">0.29</td><td align="center">0.852</td></tr><tr><td align="center">SA-TCN-LSTM<sup>[<a href="#b47" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b47">47</a>]</sup></td><td align="center">0.295</td><td align="center">0.087</td><td align="center">0.196</td><td align="center">0.909</td></tr><tr><td style="class:table_bottom_border" align="center">Proposed model</td><td style="class:table_bottom_border" align="center">0.119</td><td style="class:table_bottom_border" align="center">0.014</td><td style="class:table_bottom_border" align="center">0.095</td><td style="class:table_bottom_border" align="center">0.985</td></tr></tbody></table></div><div class="table_footer"><div class="table_footer_note"><p class="para">RMSE: Root mean square error; MSE: mean square error; MAE: mean absolute error; TCN: temporal convolutional network; GRU: gated recurrent unit; CNN: convolutional neural network; BiGRU: bidirectional gated recurrent unit; LSTM: long short-term memory; SA: self-attention.</p></div></div></div><p class="">From <a href="#Table5" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Table5">Table 5</a>, the proposed model has a lower prediction error than the other models. Consequently, the RMSE decreased by 66.1% to 73.4%, the MSE by 88.6% to 93%, and the MAE by 57.4% to 72.9%. Conversely, the <inline-formula><tex-math id="M97">$$ R^{2} $$</tex-math></inline-formula> increased by 8.36% to 24.7%. This indicates that the MICPO-SA-BiTCN-BiGRU model has a suitable architecture and accurately predicts the evolutionary trend of corrugation.</p><p class="">To discover the evolutionary trend of corrugation more intuitively, a visualization from the comparative experiment is shown in <a href="#Figure11" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure11">Figure 11</a>.</p><div class="Figure-block" id="Figure11"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2024.20/image/Figure11" class="Article-img" alt="" target="_blank"><img alt="Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU" src="https://image.oaes.cc/80d245c1-6593-4d42-8e24-7a9b32dfa757/ir4020-11.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 11. Model comparison experiment: prediction results of evolution trend of rail corrugation.</p></div></div><p class=""><a href="#Figure11" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure11">Figure 11</a> shows that the development of corrugation damage on the measured road section exists in relatively evident stages. Therefore, we divided the data collected from measurements 1 to 40 into the early stage of rail corrugation evolution, during which the CHI value increased by approximately 3.4, indicating rapid development. The data from measurements 41 to 85 were categorized as the middle stage of rail corrugation development. During this period, the CHI value increased by approximately 0.6, with the development of corrugation leveling off and fluctuations rising slowly. This indicates that the damage caused by corrugation to the rail began to intensify, and the rail was approaching a critical state. The data from measurements 86 to 98, categorized as the late stage of corrugation development, showed that the CHI value increased by approximately 1.6. During this period, the degree of corrugation damage deterioration showed a sudden increase, indicating a sharp decline in the health of the rail within a short period, thus necessitating prompt measures to curb its development.</p><p class="">Overall, the prediction trends of the models were similar; however, the proposed model was the most accurate for local prediction. In particular, during the early and late phases of corrugation damage, the model effectively captured the evolution trend of rail corrugation damage, with its predicted value closely aligned with the real value.</p></div></div></div><div id="s4" class="article-Section"><h2 >4. DISCUSSION</h2><p class="">Predicting the evolutionary trend of rail corrugation is critical for the safe operation and maintenance of railways. To address the difficulties involved in accurately evaluating the evolution state of corrugation, a method was proposed to predict the evolution trend of corrugation. By analyzing the existing on-site data on rail corrugation, the CHI and SA-BiTCN-BiGRU hybrid network models were constructed to predict the evolution process of corrugation in the time dimension. The results were better than those of existing studies.</p><p class="">However, constructing the corrugated CHI partly relies on manual experience, which is highly subjective and results in limited accuracy and standardization. In future studies, we will attempt to combine multi-source data such as on-site rail corrugation images, vibrations, and profile data to predict the evolution trend of rail corrugation. The proposed method improves the generality and reliability of our study by combining more comprehensive corrugation damage information, ensuring the safe operation of the corresponding railway line.</p><p class="">Further, we recognize the importance of predicting the location and duration of rail corrugations. Yet, the proposed method was not effective in predicting the location of rail corrugation, and the collected dataset made the prediction of duration challenging. In fault prediction and health management, most existing research focuses on predicting the development and evolution of rail corrugation, with significantly few studies addressing its location. Nevertheless, numerous scholars have studied the detection of rail corrugation positions. For instance, Yang <i>et al</i>. proposed an intelligent real-time detection method for rail corrugation using machine vision and CNN; Li <i>et al</i>. proposed an intelligent detection method for rail corrugation using signal decomposition and the entropy theory<sup>[<a href="#b48" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b48">48</a>,<a href="#b49" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b49">49</a>]</sup>. In our future work, we will aim to combine spatial data to predict the location of rail corrugation and detect rail damage promptly. Additionally, we collected annotated data on the timing and duration of rail corrugation, which can assist in predicting its duration. These efforts will significantly improve the depth of our research and represent an important direction for future studies.</p></div><div id="s5" class="article-Section"><h2 >5. CONCLUSIONS</h2><p class="">In this study, we proposed an intelligent prediction method for the evolutionary trend of rail corrugation based on SA, BiTCN, and BiGRU.</p><p class="">First, a health indicator reflecting the evolution state of the corrugation was obtained using the defined method for constructing corrugation CHI. The experimental results validated the effectiveness of the CHI. Second, we effectively demonstrated the interpretability and predictive ability of the proposed bidirectional hybrid network, SA-BiTCN-BiGRU, through an ablation experiment. Third, by using the MICPO algorithm, the optimal values of the key hyperparameters of the SA-BiTCN-BiGRU model were determined, thereby improving the prediction accuracy of the corrugation evolution trend. The findings demonstrated the high convergence capabilities of the MICPO algorithm compared to other swarm intelligent optimization algorithms. The ablation experiment strongly verified the positive role of the MICPO algorithm in improving model prediction results. Finally, the results of the model comparison confirmed that the MICPO-SA-BiTCN-BiGRU model is efficient. The proposed method is significant for railway maintenance, as it effectively predicts the future development trend of rail corrugation and provides a scientific basis for railway maintenance decisions.</p></div><div id="s6" class="article-Section"><h2 >DECLARATIONS</h2><div id="s6-1" class="article-Section"><h3 >Acknowledgments</h3><p class="">We thank the Editor-in-Chief and all reviewers for their comments.</p></div><div id="s6-2" class="article-Section"><h3 >Authors' contributions</h3><p class="">Conducted experimental analysis and manuscript writing: Yang WH</p><p class="">Guided on the overall framework and implementation steps of this research, and proposed a train of thought for the general research objectives: Liu JH, Zhang CF</p><p class="">Guided English writing: He J</p><p class="">Provided technical support: Wang ZM, Jia L</p><p class="">Provided dataset support: Yang WW</p></div><div id="s6-3" class="article-Section"><h3 >Availability of data and materials</h3><p class="">The data are available upon request. If needed, please contact the corresponding author by email.</p></div><div id="s6-4" class="article-Section"><h3 >Financial support and sponsorship</h3><p class="">This research was funded by the National Key Research and Development Program (Grant No. 2021YFF0501101), the National Natural Science Foundation of China (Grant Nos. 52272347, 62303178), Key Scientific Research Project of the Hunan Provincial Department of Education (Grant No. 22A0391), the Natural Science Foundation of the Hunan Province (Grant No. 2024JJ7132).</p></div><div id="s6-5" class="article-Section"><h3 >Conflicts of interest</h3><p class="">Yang WW is affiliated with Zhuzhou Qingyun Electric Locomotive Accessories Factory Co., Ltd., while the other authors have declared that they have no conflicts of interest.</p></div><div id="s6-6" class="article-Section"><h3 >Ethical approval and consent to participate</h3><p class="">Not applicable.</p></div><div id="s6-7" class="article-Section"><h3 >Consent for publication</h3><p class="">Not applicable.</p></div><div id="s6-8" class="article-Section"><h3 >Copyright</h3><p class="">© The Author(s) 2024.</p></div></div></div>  <div class="art_list" data-v-6dffe839></div> <div class="article_references" data-v-6dffe839><div class="ReferencesBox" data-v-6dffe839><h2 id="References" class="bg_d" data-v-6dffe839><span data-v-6dffe839><a href="/articles//reference" data-v-6dffe839>REFERENCES</a></span> <span class="icon" data-v-6dffe839><i class="el-icon-arrow-down" data-v-6dffe839></i> <i class="el-icon-arrow-up hidden" data-v-6dffe839></i></span></h2> <div class="references_list heightHide" data-v-6dffe839><div id="b1" class="references_item" data-v-6dffe839><p data-v-6dffe839><span data-v-6dffe839>1. </span> <span data-v-6dffe839>Wang Z, Lei Z. 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Intelligent detection of rail corrugation using ACMP-based energy entropy and LSSVM. <i>Nonlin Dynam</i> 2023;111:8419-38.</span></p> <div class="refrences" data-v-6dffe839><a href="https://dx.doi.org/10.1007/s11071-022-08066-2" target="_blank" data-v-6dffe839><button type="button" class="el-button el-button--text el-button--mini" data-v-6dffe839><span>DOI</span></button></a>  </div></div></div> <div class="line" data-v-6dffe839></div></div> <div class="article_cite cite_layout" data-v-6dffe839><div id="cite" data-v-6dffe839></div> <div class="el-row" style="margin-left:-10px;margin-right:-10px;" data-v-6dffe839><div class="el-col el-col-24 el-col-xs-24 el-col-sm-16" style="padding-left:10px;padding-right:10px;" data-v-6dffe839><div class="left_box" data-v-6dffe839><div data-v-6dffe839><h2 style="margin-top:0!important;padding-top:0;" data-v-6dffe839>Cite This Article</h2> <div class="cite_article" data-v-6dffe839><div class="cite_article_sec" data-v-6dffe839>Research Article</div> <div class="cite_article_open" style="color:#aa0c2f;" data-v-6dffe839><img src="https://g.oaes.cc/oae/nuxt/img/open_icon.bff5dde.png" alt="" style="width:10px;" data-v-6dffe839> Open Access</div> <div class="cite_article_tit" data-v-6dffe839><span data-v-6dffe839>Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU</span></div> <div class="cite_article_editor" data-v-6dffe839><span data-v-6dffe839>Jian-Hua Liu, ... 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src="data:image/png;base64,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{layout:"oaelayouta",data:[{ArtData:{date_published:"2024-10-18 00:00:00",section:t,title:bF,doi:"10.20517\u002Fir.2024.20",abstract:bG,pdfurl:"https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020.pdf",xmlurl:"https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020.xml",elocation_id:b,fpage:318,article_id:e,viewed:d,downloaded:d,video_url:bH,volume:f,year:q,cited:d,corresponding:"Correspondence to: Dr. Zhong-Mei Wang, College of Railway Transportation, Hunan University of Technology, No 88, Taishan\r\nWest Road, Taiyuan District, Zhuzhou 412007, Hunan, China. E-mail: \u003Cemail\u003Ewangzhongmei@hut.edu.cn\u003C\u002Femail\u003E",editor:[],editor_time:"\u003Cspan\u003E\u003Cb\u003EReceived:\u003C\u002Fb\u003E 28 Aug 2024 | \u003C\u002Fspan\u003E\u003Cspan\u003E\u003Cb\u003EFirst Decision:\u003C\u002Fb\u003E 14 Sep 2024 | \u003C\u002Fspan\u003E\u003Cspan\u003E\u003Cb\u003ERevised:\u003C\u002Fb\u003E 21 Sep 2024 | \u003C\u002Fspan\u003E\u003Cspan\u003E\u003Cb\u003EAccepted:\u003C\u002Fb\u003E 11 Oct 2024 | \u003C\u002Fspan\u003E\u003Cspan\u003E\u003Cb\u003EPublished:\u003C\u002Fb\u003E 18 Oct 2024\u003C\u002Fspan\u003E",cop_link:"https:\u002F\u002Fcreativecommons.org\u002Flicenses\u002Fby\u002F4.0\u002F",cop_info:"© The Author(s) 2024. \u003Cb\u003EOpen Access\u003C\u002Fb\u003E This article is licensed under a Creative Commons Attribution 4.0 International License (\u003Ca target=\"_blank\"href=\"https:\u002F\u002Fcreativecommons.org\u002Flicenses\u002Fby\u002F4.0\u002F\" xmlns:xlink=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxlink\"\u003Ehttps:\u002F\u002Fcreativecommons.org\u002Flicenses\u002Fby\u002F4.0\u002F\u003C\u002Fa\u003E), which permits unrestricted use, sharing, adaptation, distribution and reproduction in any medium or format, for any purpose, even commercially, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.",keywords:["Mahalanobis distance","rail corrugation","evolution trend prediction","improved crested porcupine optimizer","hybrid time series network"],issue:f,image:r,tag:" \u003Ci\u003EIntell Robot\u003C\u002Fi\u003E 2024;4(4):318-38.",authors:"Jian-Hua Liu, ... Wei-Wei Yang",picurl:b,expicurl:b,picabstract:b,interview_pic:a,interview_url:b,review:a,cop_statement:"© The Author(s) 2024.",seo:[],video_img:bI,lpage:338,author:[{base:"Jian-Hua Liu\u003Csup\u003E1\u003C\u002Fsup\u003E",email:a,orcid:a},{base:"Wei-Hao Yang\u003Csup\u003E1\u003C\u002Fsup\u003E",email:a,orcid:a},{base:"Jing He\u003Csup\u003E2\u003C\u002Fsup\u003E",email:a,orcid:a},{base:"Zhong-Mei Wang\u003Csup\u003E1\u003C\u002Fsup\u003E",email:"wangzhongmei@hut.edu.cn",orcid:a},{base:"Lin Jia\u003Csup\u003E1\u003C\u002Fsup\u003E",email:a,orcid:a},{base:"Chang-Fan Zhang\u003Csup\u003E1\u003C\u002Fsup\u003E",email:a,orcid:a},{base:"Wei-Wei Yang\u003Csup\u003E3\u003C\u002Fsup\u003E",email:a,orcid:a}],specialissue:a,specialinfo:a,date_published_stamp:1729180800,year1:q,CitedImage:"https:\u002F\u002Fi.oaes.cc\u002Fimages_2018\u002Fjournals\u002FCrossref.png",article_editor:[],editoruser:"\u003Cspan\u003E\u003Cb\u003EAcademic Editor:\u003C\u002Fb\u003E Simon Yang | \u003C\u002Fspan\u003E\u003Cspan\u003E\u003Cb\u003ECopy Editor:\u003C\u002Fb\u003E Pei-Yun Wang | \u003C\u002Fspan\u003E\u003Cspan\u003E\u003Cb\u003EProduction Editor:\u003C\u002Fb\u003E Pei-Yun Wang\u003C\u002Fspan\u003E",commentsNums:d,oaestyle:bJ,amastyle:bK,ctstyle:bL,acstyle:bM,copyImage:"https:\u002F\u002Fi.oaes.cc\u002Fimages_2018\u002Fjournals\u002Fccb_4.png",affiliation:[{id:97450,article_id:e,Content:"\u003Clabel\u003E1\u003C\u002Flabel\u003E\u003Caddr-line\u003ECollege of Railway Transportation, Hunan University of Technology, Zhuzhou 412007, Hunan, China.\u003C\u002Faddr-line\u003E"},{id:97451,article_id:e,Content:"\u003Clabel\u003E2\u003C\u002Flabel\u003E\u003Caddr-line\u003ECollege of Electrical and Information Engineering, Hunan University of Technology, Zhuzhou 412007, Hunan, China.\u003C\u002Faddr-line\u003E"},{id:97452,article_id:e,Content:"\u003Clabel\u003E3\u003C\u002Flabel\u003E\u003Caddr-line\u003EZhuzhou Qingyun Electric Locomotive Accessories Factory Co., Ltd, Zhuzhou 412007, Hunan, China.\u003C\u002Faddr-line\u003E"}],related:[],down:"https:\u002F\u002Ff.oaes.cc\u002Fris\u002F7277.ris",xml:{id:5358,article_id:e,xml_down:d,cite_click:d,export_click:d},zan:d,cited_type:"cited",subarray:[],issn:bh,uuid:"ec396ae3b161ca1841daa46a9becc7b7",abstractUuid:"7892ee631927e24428f33adaf05cd33d",apiurl:a,api_abstract_url:a,journal_id:at,journal_path:N},loadingAbs:void 0,loading:O,ArtDataC:{content:"\u003Cdiv id=\"s1\" class=\"article-Section\"\u003E\u003Ch2 \u003E1. INTRODUCTION\u003C\u002Fh2\u003E\u003Cp class=\"\"\u003ELong-term wheel-rail contact on railway lines can cause various types of damage, particularly in sections with small curvature radius, where corrugation damage is more prevalent\u003Csup\u003E[\u003Ca href=\"#b1\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b1\"\u003E1\u003C\u002Fa\u003E-\u003Ca href=\"#b3\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b3\"\u003E3\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. Rail corrugation primarily affects the inner surface of the rail in a curved section, resulting in periodic wavy wear. If left undetected and unrepaired, corrugation can cause train vibrations, significantly reducing its operating stability. In severe cases, rail breakage and major accidents, such as train derailments, can also occur\u003Csup\u003E[\u003Ca href=\"#b4\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b4\"\u003E4\u003C\u002Fa\u003E,\u003Ca href=\"#b5\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b5\"\u003E5\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. Therefore, in railway health management, in-depth research on the evolution of rail corrugation is critical\u003Csup\u003E[\u003Ca href=\"#b6\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b6\"\u003E6\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E to ensuring the safe operation of rail transit\u003Csup\u003E[\u003Ca href=\"#b7\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b7\"\u003E7\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EOver the past years, scholars have conducted in-depth research on the generation and evolution process of rail corrugation, mainly using two methods: mechanism modeling and data-driven prediction. In mechanism modeling, the wheel-rail transient dynamics method is used to establish a model that reflects the evolution process of corrugation. Additionally, by using mechanical simulation software, scholars have constructed wheel-rail coupling finite element models and rail elastic-plastic analysis models to further explore the generation and evolution\u003Csup\u003E[\u003Ca href=\"#b8\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b8\"\u003E8\u003C\u002Fa\u003E,\u003Ca href=\"#b9\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b9\"\u003E9\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E of corrugation. For example, Wang \u003Ci\u003Eet al\u003C\u002Fi\u003E. established a vehicle-track space coupling model using multibody dynamics software and conducted a dynamic analysis of the corrugation section\u003Csup\u003E[\u003Ca href=\"#b1\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b1\"\u003E1\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. Cui \u003Ci\u003Eet al\u003C\u002Fi\u003E. established a finite element model of the wheel-rail system and a wear model for corrugation using typical rail corrugation on a curve with a small radius as the research object; they then elucidated the development mechanism of corrugation by studying the dynamic response of the wheel-rail on the rail surface\u003Csup\u003E[\u003Ca href=\"#b2\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b2\"\u003E2\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. However, these methods rely on prior knowledge of factors, such as the damage mechanism, and are highly theoretical. Furthermore, achieving an optimal damage evolution process using these methods in a complex train operating environment is challenging.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EIn data-driven research, scholars typically use experimental or on-site data to extract damage degradation features. Machine and deep learning methods are employed for damage diagnosis or prediction tasks without requiring an in-depth understanding of the internal damage mechanisms, as these methods can indirectly consider various influencing factors\u003Csup\u003E[\u003Ca href=\"#b10\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b10\"\u003E10\u003C\u002Fa\u003E-\u003Ca href=\"#b13\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b13\"\u003E13\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. For example, Xiao \u003Ci\u003Eet al\u003C\u002Fi\u003E. used machine learning to detect and assess corrugation damage in heavy haul railways; their approach, which was based on support vector machines and other technologies, could effectively detect rail corrugation damage\u003Csup\u003E[\u003Ca href=\"#b14\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b14\"\u003E14\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. Deep network models such as gated recurrent units (GRUs), temporal convolutional networks (TCNs), and attention mechanisms are widely used in industrial equipment for damage diagnosis and degradation trend prediction\u003Csup\u003E[\u003Ca href=\"#b15\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b15\"\u003E15\u003C\u002Fa\u003E-\u003Ca href=\"#b20\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b20\"\u003E20\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E because of their exceptional feature extraction and nonlinear mapping abilities. For example, Zhang \u003Ci\u003Eet al\u003C\u002Fi\u003E. introduced a squeeze-excitation channel attention mechanism into a combined model of a convolutional neural network (CNN) and bidirectional GRU (BiGRU); this integration demonstrated that the addition of an attention mechanism improved the capability of the network to focus on excellent features\u003Csup\u003E[\u003Ca href=\"#b21\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b21\"\u003E21\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. Liu \u003Ci\u003Eet al\u003C\u002Fi\u003E. used a dynamic multiscale gated causal convolution method combined with a GRU to effectively predict the actual degradation trend of rail corrugation and address poor generalization caused by small data samples\u003Csup\u003E[\u003Ca href=\"#b22\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b22\"\u003E22\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. Additionally, in the general damage evolution prediction task, degradation is a continuous change process with a front-back relationship over time\u003Csup\u003E[\u003Ca href=\"#b23\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b23\"\u003E23\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. Currently, most scholars do not consider the relationship between the time series before and after the damage signal. In a complex time-series prediction task, a single model often has limitations in terms of generalization, robustness, and adaptability. The current hybrid temporal prediction networks typically rely on extensive experiments and parameter-tuning processes, thereby increasing the computational cost and making the optimality of the selected hyperparameters difficult.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003ETo address the aforementioned shortcomings, this study constructed a self-attention (SA) bidirectional TCN and GRU (SA-BiTCN-BiGRU) hybrid network and used a new multi-strategy improved crested porcupine optimizer (MICPO) algorithm for automatic hyperparameter optimization. The proposed model integrated the advantages of each module, exhibiting robust time-series modeling capabilities, perceiving dynamic changes in a time series, and assigning more weight to important time-series features. Thus, the prediction accuracy of the evolution trend of rail corrugation improved. The MICPO algorithm could automatically determine the optimal network hyperparameters for the proposed network using the four improvement strategies and its superior global search ability, thereby enhancing the network's prediction accuracy and reducing the need for blind manual adjustment of hyperparameters. Finally, the efficacy and superiority of the proposed methodology were verified through experiments and compared with other advanced methods.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EThe remainder of this study is organized as follows. Section 2 introduces the construction method of the rail corrugation's comprehensive health indicator (CHI), corrugation evolution trend prediction model, and model hyperparameter optimization algorithm. Section 3 describes the experimental setup and preprocessing of the rail corrugation dataset, and subsequently analyzes the experimental results in detail. Section 4 provides a comprehensive summary of the research content and proposes current limitations and future research directions. Finally, Section 5 concludes the study.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s2\" class=\"article-Section\"\u003E\u003Ch2 \u003E2. METHODS\u003C\u002Fh2\u003E\u003Cp class=\"\"\u003EBased on the current research background, this section provides a detailed description of the process for predicting the evolutionary trend of rail corrugation. A corrugation CHI was established using the collected on-site dataset. A SA-BiTCN-BiGRU hybrid network was used to predict the evolution trend of rail corrugation, and the MICPO algorithm was constructed to adaptively adjust the hyperparameters of the network. The overall framework is illustrated in \u003Ca href=\"#Figure1\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure1\"\u003EFigure 1\u003C\u002Fa\u003E.\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure1\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure1\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU\" src=\"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-1.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 1. Frame diagram of rail corrugation trend prediction.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003EAs can be seen from \u003Ca href=\"#Figure1\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure1\"\u003EFigure 1\u003C\u002Fa\u003E, first, by observing the damage changes in the corrugation image, the three vibration sensors were installed at the front wheel, rear wheel, and center position of the bogie on the track inspection car to collect corrugation vibration data in the vertical direction. After preprocessing the collected data, the rail corrugation vibration signal was obtained. Subsequently, multidomain feature extraction, feature screening, feature dimensionality reduction, and the Mahalanobis distance (MD) measurement methods were applied to this corrugation vibration signal, resulting in a CHI that effectively characterized the evolution trend of rail corrugation. The CHI was then input into the SA-BiTCN-BiGRU hybrid network to predict the evolution trend of rail corrugation. The network integrated the advantages of BiTCN, BiGRU, and SA to address the limitations of existing models. Finally, the MICPO algorithm was used to accurately select the optimal network model hyperparameters, thereby effectively improving the prediction accuracy of the model.\u003C\u002Fp\u003E\u003Cdiv id=\"s2-1\" class=\"article-Section\"\u003E\u003Ch3 \u003E2.1. Collection of rail corrugation signal and construction of corrugation CHI\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003EIn this study, three vibration sensors installed on the track inspection car were used to obtain vibration data of corrugation damage from different positions in the same direction. Compared with the data of a single sensor, the multi-channel data contains richer feature information and can more comprehensively reflect the changing characteristics of corrugation damage\u003Csup\u003E[\u003Ca href=\"#b24\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b24\"\u003E24\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. Therefore, to fully explore the vibration information of the three channels, we first normalized the data of each channel to reduce the impact of the difference in signal distribution between different channels. The multi-channel signal fusion method based on kurtosis weight was then used to calculate the fusion weight of the three channels, and the signals from each channel were subjected to weighted fusion. The kurtosis value can effectively reflect the severity of rail corrugation damage. The channel with a higher kurtosis value is considered to be more sensitive to the reflection of corrugation damage, so a higher weight is assigned to ensure that more representative vibration signals have a more significant impact on the overall analysis results during the fusion process, so that the merged vibration signals can reflect the changing trend of rail corrugation damage more comprehensively and reliably\u003Csup\u003E[\u003Ca href=\"#b25\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b25\"\u003E25\u003C\u002Fa\u003E,\u003Ca href=\"#b26\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b26\"\u003E26\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EThe CHI is an indicator used to evaluate and quantify the evolution trend of rail corrugation. The construction of a CHI is a preprocessing step for predicting the evolution of corrugation, which influences the effectiveness of subsequent prediction tasks\u003Csup\u003E[\u003Ca href=\"#b27\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b27\"\u003E27\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. However, in a complex environment, various adverse factors may lead to significant deviations in the extracted rail corrugation vibration data, resulting in a lack of reliability in the constructed CHI. Therefore, this study used a custom range box line method to identify outliers in the corrugation vibration data and performed a mean correction on these outliers, thereby improving data quality. To accurately construct the CHI of corrugation and overcome the problem of relying on manual experience selection for single physical and fusion indicators, this study establishes a CHI that reflects the evolution of corrugation. The process steps are described as follows.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EFirst, the vibration data of rail corrugation collected from the field contain numerous degradation features reflecting the evolution process of corrugation. The amplitude of these features usually deviates from the normal range with time, indicating that the corrugation damage is intensifying\u003Csup\u003E[\u003Ca href=\"#b28\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b28\"\u003E28\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. Therefore, this study extracted time-domain, frequency-domain, and time-frequency domain feature indicators from the data, such as the maximum value, root mean square (RMS), standard deviation, and pulse index. These feature indicators effectively reflect corrugation degradation through the concretization of abstract real data.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003ESubsequently, three evaluation indicators, monotonicity (\u003Cinline-formula\u003E\u003Ctex-math id=\"M1\"\u003E$$ M $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E), Spearman's correlation coefficient (\u003Cinline-formula\u003E\u003Ctex-math id=\"M2\"\u003E$$ S $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E), and robustness (\u003Cinline-formula\u003E\u003Ctex-math id=\"M3\"\u003E$$ R $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E), were used to quantify the damage features of rail corrugation\u003Csup\u003E[\u003Ca href=\"#b29\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b29\"\u003E29\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. Concurrently, to conduct comprehensive evaluation of each rail's corrugation features, this study used normalization processing to quantify the three evaluation indicators to the same scale, eliminating the impact of dimension, and obtained comprehensive evaluation indicator (\u003Cinline-formula\u003E\u003Ctex-math id=\"M4\"\u003E$$ C $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E)\u003Csup\u003E[\u003Ca href=\"#b30\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b30\"\u003E30\u003C\u002Fa\u003E,\u003Ca href=\"#b31\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b31\"\u003E31\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E through linear weighted combination of the three evaluation indicators, using it to calculate the comprehensive score of each feature, and adaptively screen out the features sensitive to the change of rail's corrugation state, forming an optimal feature subset. The definitions of \u003Cinline-formula\u003E\u003Ctex-math id=\"M5\"\u003E$$ M $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, \u003Cinline-formula\u003E\u003Ctex-math id=\"M6\"\u003E$$ S $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, \u003Cinline-formula\u003E\u003Ctex-math id=\"M7\"\u003E$$ R $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, and \u003Cinline-formula\u003E\u003Ctex-math id=\"M8\"\u003E$$ C $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E are\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(1)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E1\"\u003E $$ \\begin{equation} M=\\frac{1}{T-1}\\Bigg|X\\Bigg(\\frac{d}{df_t}>0\\Bigg)-Y\\Bigg(\\frac{d}{df_t}<0\\Bigg)\\Bigg| \\\\ \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(2)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E2\"\u003E $$ \\begin{equation} S=1-\\frac{6\\sum d_t^2}{T\\left(T^2-1\\right)} \\\\ \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(3)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E3\"\u003E $$ \\begin{equation} R=\\frac{1}{T}\\sum\\limits_{t=1}^{T}\\exp\\left(-\\left|\\frac{f_{t}-\\tilde{f}_{t}}{f_{t}}\\right|\\right) \\\\ \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(4)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E4\"\u003E $$ \\begin{equation} C=\\frac{M+S+R}{3} \\\\ \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003Erespectively, where \u003Cinline-formula\u003E\u003Ctex-math id=\"M9\"\u003E$$ T $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E indicates the length of the degradation feature sequence, \u003Cinline-formula\u003E\u003Ctex-math id=\"M10\"\u003E$$ f_{t} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is the extracted value of the feature at time \u003Cinline-formula\u003E\u003Ctex-math id=\"M11\"\u003E$$ t $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, \u003Cinline-formula\u003E\u003Ctex-math id=\"M12\"\u003E$$ X(\\cdot) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E denotes the number of positive derivatives in the degradation feature sequence, \u003Cinline-formula\u003E\u003Ctex-math id=\"M13\"\u003E$$ Y(\\cdot) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E denotes the number of negative derivatives in the degradation feature sequence, \u003Cinline-formula\u003E\u003Ctex-math id=\"M14\"\u003E$$ \\Sigma $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E denotes the summation symbol, \u003Cinline-formula\u003E\u003Ctex-math id=\"M15\"\u003E$$ d_{t} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is the difference between the degradation feature index sequence and time series, \u003Cinline-formula\u003E\u003Ctex-math id=\"M16\"\u003E$$ \\exp(\\cdot) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E represents an exponential function based on the natural constant \u003Cinline-formula\u003E\u003Ctex-math id=\"M17\"\u003E$$ e $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, and \u003Cinline-formula\u003E\u003Ctex-math id=\"M18\"\u003E$$ \\tilde{f}_t $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is the value of the feature at time \u003Cinline-formula\u003E\u003Ctex-math id=\"M19\"\u003E$$ t $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E after sliding average.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EPrincipal component analysis (PCA) was then used to reduce the dimensionality of the optimal feature subset, and the principal components were weighted according to their contribution degrees to generate a multidimensional principal component vector that retains the important information of the original optimal feature subset.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EFinally, the MD\u003Csup\u003E[\u003Ca href=\"#b32\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b32\"\u003E32\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E was used to calculate the difference between the initial and subsequent samples in the generated multidimensional principal component vector. The obtained results were then smoothed using the exponential weighted moving average, yielding the CHI, which reflected the evolution of corrugation. The MD is calculated as follows:\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(5)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E5\"\u003E $$ \\begin{equation} D_M\\left(m,n\\right)=\\sqrt{\\left(m-n\\right)^T\\text{K}^{-1}\\left(m-n\\right)} \\\\ \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003Ewhere \u003Cinline-formula\u003E\u003Ctex-math id=\"M20\"\u003E$$ m $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M21\"\u003E$$ n $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E are the sample vectors; \u003Cinline-formula\u003E\u003Ctex-math id=\"M22\"\u003E$$ K $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E represents the covariance matrix of the corrugation evolution features.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s2-2\" class=\"article-Section\"\u003E\u003Ch3 \u003E2.2. Establishment of trend prediction model for rail corrugation\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003ETo accurately predict the evolution trend of rail corrugation, we constructed a SA-BiTCN-BiGRU model. Using the initial corrugation data in the established CHI as the input, the subsequent CHI values were predicted.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EThe structure of the model is illustrated in \u003Ca href=\"#Figure2\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure2\"\u003EFigure 2\u003C\u002Fa\u003E. First, the bidirectional local features of the initial corrugation data were effectively extracted using a three-layer BiTCN to improve the receptive field and feature extraction capability of the model. Subsequently, based on the local features extracted by BiTCN, BiGRU was used for time-series prediction, and the output results were passed through the Leaky rectified linear unit (ReLU) nonlinear activation function and dropout regularization technology. The attention weight provided by the SA was then used to enhance the interpretability of the network. Finally, the multilayer perceptron (MLP) network output continuous prediction results and the error between them and the actual value was calculated to evaluate the prediction effect of the model.\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure2\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure2\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU\" src=\"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-2.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 2. Architecture of prediction model for rail corrugation trend.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s2-2-1\" class=\"article-Section\"\u003E\u003Ch4 \u003E2.2.1. BiTCN\u003C\u002Fh4\u003E\u003Cp class=\"\"\u003EIn this study, the constructed corrugation CHI is a continuous process that changes over time, and the data are closely related. To capture the features of the corrugation CHI over a wider range, a BiTCN was constructed to comprehensively consider the historical and forthcoming temporal information of the corrugation CHI. Additionally, multiple dilated causal convolution layers were stacked to improve the receptive field, effectively observe the change patterns in the rail corrugation data, and enhance the model's capacity to acquire key information. The structure of the BiTCN is shown in \u003Ca href=\"#Figure3\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure3\"\u003EFigure 3\u003C\u002Fa\u003E.\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure3\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure3\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU\" src=\"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-3.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 3. Schematic of BiTCN structure. BiTCN: Bidirectional temporal convolutional network.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003EAs shown in \u003Ca href=\"#Figure3\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure3\"\u003EFigure 3\u003C\u002Fa\u003E, BiTCN consists of a forward and a reverse TCN residual block linked together. The model's output was the combined training result of the two blocks. Each residual block contained two layers of dilated causal convolution, which enlarged the receptive field of the network. The input sequence data were derived from the one-dimensional rail corrugation CHI, and the feature information at different scales was captured using the dilated convolution operation. A batch normalization layer was employed to stabilize the model training process. The Leaky ReLU activation function enabled the BiTCN module to train a deeper network while addressing dead neurons and vanishing gradient. Additionally, dropout regularization technology was added to reduce overfitting. To accommodate possible differences in the number of input and output channels in the model, a 1 × 1 convolution layer was added in each training direction for the residual connection, and the number of feature channels was adjusted to suit the feature representations of different levels.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EThe core concept of the dilated causal convolution involves the insertion of zero elements into the convolution kernel, which modifies the structure of the kernel and effectively expands the receptive field of the model. This enables each convolution output to encompass a broader range of time information, effectively mitigating vanishing gradient caused by numerous layers in the common convolution and enabling the model to extract more information on corrugation evolution\u003Csup\u003E[\u003Ca href=\"#b33\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b33\"\u003E33\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. The internal structure of the dilated causal convolution is shown in \u003Ca href=\"#Figure4\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure4\"\u003EFigure 4\u003C\u002Fa\u003E and defined below:\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(6)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E6\"\u003E $$ \\begin{equation} y=\\sum\\limits_{k=0}^{K-1}\\omega\\bigl[k\\bigr]\\cdot x\\bigl[L-d\\cdot k\\bigr] \\\\ \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure4\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure4\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU\" src=\"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-4.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 4. Visualization of dilated causal convolution.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003Ewhere \u003Cinline-formula\u003E\u003Ctex-math id=\"M23\"\u003E$$ y $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is the output of the dilated causal convolution layer, \u003Cinline-formula\u003E\u003Ctex-math id=\"M24\"\u003E$$ \\omega[k] $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E denotes the weight of the convolution kernel \u003Cinline-formula\u003E\u003Ctex-math id=\"M25\"\u003E$$ k $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, \u003Cinline-formula\u003E\u003Ctex-math id=\"M26\"\u003E$$ x[L-d\\cdot k] $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E denotes the value of the input sequence element, \u003Cinline-formula\u003E\u003Ctex-math id=\"M27\"\u003E$$ L $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is the length of the input sequence, and \u003Cinline-formula\u003E\u003Ctex-math id=\"M28\"\u003E$$ d $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E represents the expansion rate.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s2-2-2\" class=\"article-Section\"\u003E\u003Ch4 \u003E2.2.2. BiGRU\u003C\u002Fh4\u003E\u003Cp class=\"\"\u003EThe GRU is a temporal prediction network proposed to alleviate the vanishing gradient problem of a recurrent neural network (RNN)\u003Csup\u003E[\u003Ca href=\"#b34\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b34\"\u003E34\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. The evolution of corrugation is closely related to information from past and future data, and unidirectional GRU may fail to capture this bidirectional information transmission mode. Therefore, this study constructed a BiGRU to infer the relationship between past and future corrugation characteristics and the current corrugation amplitude to improve the model's sensitivity and predictive capability regarding dynamic changes in the time series of corrugation characteristics. The BiGRU is calculated using\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(7)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E7\"\u003E $$ \\begin{equation} \\begin{cases}\\overrightarrow{h_n}=GRU\\left(x_n,\\overrightarrow{h}_{n-1}\\right)\\\\\\\\\\overleftarrow{h_n}=GRU\\left(x_n,\\overleftarrow{h}_{n-1}\\right)\\\\\\\\h_n=\\alpha_n\\overrightarrow{h}_n+\\beta_n\\overleftarrow{h}_n+b_n&\\end{cases} \\\\ \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003Ewhere \u003Cinline-formula\u003E\u003Ctex-math id=\"M29\"\u003E$$ GRU(\\cdot) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E denotes the gated cycle unit, \u003Cinline-formula\u003E\u003Ctex-math id=\"M30\"\u003E$$ x_{n} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is the input, \u003Cinline-formula\u003E\u003Ctex-math id=\"M31\"\u003E$$ \\overrightarrow{h_n} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M32\"\u003E$$ \\overleftarrow{h_n} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E represent the output status of the forward and reverse hidden layers, respectively, \u003Cinline-formula\u003E\u003Ctex-math id=\"M33\"\u003E$$ \\alpha_{n} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M34\"\u003E$$ \\beta_{n} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E are the corresponding output weights, and \u003Cinline-formula\u003E\u003Ctex-math id=\"M35\"\u003E$$ b_{n} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is the corresponding bias. The structure of the BiGRU is shown in \u003Ca href=\"#Figure5\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure5\"\u003EFigure 5\u003C\u002Fa\u003E.\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure5\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure5\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU\" src=\"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-5.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 5. Schematic of BiTCN structure. BiTCN: Bidirectional temporal convolutional network.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003EThe entire network is composed of an input layer, two layers of GRUs in opposite directions, and an output layer. The input is the value after BiTCN feature extraction, and the output is determined based on the cycling training results of the BiGRU unit.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s2-2-3\" class=\"article-Section\"\u003E\u003Ch4 \u003E2.2.3. SA mechanism\u003C\u002Fh4\u003E\u003Cp class=\"\"\u003EAs a variant of the attention mechanism, SA\u003Csup\u003E[\u003Ca href=\"#b35\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b35\"\u003E35\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E is mainly used to process serial data such as the rail corrugation time-series data used in this study. The network can calculate the attention weights of various positions at different time steps to improve its ability to obtain key information and integrate the content of all time steps. This study introduced and applied the SA mechanism to the process of model trend prediction, which was designed to improve the model's dependence on different locations in the input ripple CHI sequence. This allowed the model to better understand the internal correlations between the corrugation data at each moment, significantly improving its predictive performance. This technology can provide reliable decision-making support for the maintenance and management of railway systems. \u003Ca href=\"#Figure6\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure6\"\u003EFigure 6\u003C\u002Fa\u003E shows the structure of the SA.\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure6\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure6\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU\" src=\"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-6.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 6. Illustration of SA mechanism. SA: Self-attention.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003EAs depicted in \u003Ca href=\"#Figure6\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure6\"\u003EFigure 6\u003C\u002Fa\u003E, the structure initially computes and packages the query, key, and value vectors of all input matrices as matrices. The query and key vectors were used to perform a nonlinear transformation. The dot product and masking operations standardized the query and key vectors, masked invalid information, and generated an attention score. The mapping matrix of the attention score was then obtained after normalization using the softmax operation and multiplied by the value vector after identity mapping to acquire the weight output.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s2-3\" class=\"article-Section\"\u003E\u003Ch3 \u003E2.3. Model hyperparameter optimization based on MICPO algorithm\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003ECertain hyperparameters significantly affected the predictive performance of the proposed model. For example, the convolution kernel size determined the capability of the model to capture corrugation characteristics in the time dimension. The number of BiGRU hidden layer units determines the complexity and learning ability of the network. To prevent the adverse effects of manual intervention in the selection of model hyperparameters, optimization algorithms are necessary to adaptively identify the most suitable model hyperparameters.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EConsequently, model hyperparameter optimization was performed using the crested porcupine optimizer (CPO)\u003Csup\u003E[\u003Ca href=\"#b36\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b36\"\u003E36\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E algorithm. This algorithm simulates four different defense strategies when a crested porcupine (CP) engages in defense against predators. The first two strategies, sight and sound, represent the exploration phase of the algorithm; the last two strategies, odor and physical-attack, represent the exploitation phase of the algorithm. Different defense strategies have distinct optimization effects on various hyperparameters, guiding the algorithm to identify the optimal hyperparameters for the model. However, the original algorithm has certain limitations, such as decreasing population diversity and the tendency to get trapped in local optimality in the later stages of a search, leading to an inaccurate selection of hyperparameters. Therefore, a multi-strategy improvement method was constructed to optimize the initialization mode and defense strategy of the CPO algorithm to acquire better model hyperparameters and enhance the prediction accuracy of the model on the evolution trend of rail corrugation. The detailed improvement strategies for the CPO algorithm are discussed in the following subsections.\u003C\u002Fp\u003E\u003Cdiv id=\"s2-3-1\" class=\"article-Section\"\u003E\u003Ch4 \u003E2.3.1. Improved tent chaos map\u003C\u002Fh4\u003E\u003Cp class=\"\"\u003EIn the algorithm initialization stage, an improved tent map was employed to generate chaotic sequences and address issues related to the reduction of the CP population and its tendency to converge into the local optimal solution when the CPO algorithm approached the global optimum\u003Csup\u003E[\u003Ca href=\"#b37\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b37\"\u003E37\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. This method introduced random variables into a traditional tent-chaos map. Thus, the diversity of the CP individuals was increased, and the chaotic sequence was prevented from falling into unstable periodic points during the iterative process defined as follows:\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(8)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E8\"\u003E $$ \\begin{equation} X_{i,j+1}=\\begin{cases}\\frac{X_{i,j}}{tent}+rand\\left(0,1\\right),&\\quad0\\leq{X}_{i,j}\\leq tent\\\\\\\\\\frac{1-X_{i,j}}{1-tent}+rand\\left(0,1\\right),&\\quad{tent}<{X}_{i,j}\\leq1\\end{cases} \\\\ \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003Ewhere \u003Cinline-formula\u003E\u003Ctex-math id=\"M36\"\u003E$$ i $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M37\"\u003E$$ j $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E represent the CP population number and current dimension, respectively; \u003Cinline-formula\u003E\u003Ctex-math id=\"M38\"\u003E$$ tent $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E denotes the chaos coefficient, and \u003Cinline-formula\u003E\u003Ctex-math id=\"M39\"\u003E$$ rand\\begin{pmatrix}0,1\\end{pmatrix} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E represents a random number between 0 and 1.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s2-3-2\" class=\"article-Section\"\u003E\u003Ch4 \u003E2.3.2. Golden sine strategy\u003C\u002Fh4\u003E\u003Cp class=\"\"\u003EIn this study, the golden sine strategy\u003Csup\u003E[\u003Ca href=\"#b38\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b38\"\u003E38\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E was incorporated into the CPO algorithm to enlarge its search space and address the lack of information exchange between CP individuals in the original algorithm, thereby improving the algorithm's ability for global optimization defined as follows:\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(9)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E9\"\u003E $$ \\begin{equation} X_{i,j}^{t+1}=X_{i,j}^{t}\\times\\left|\\sin\\left(D_{1}\\right)\\right|+D_{2}\\times\\sin\\left(D_{1}\\right)\\times\\left|x_{1}\\times X_{i,j}^{t}-x_{2}\\times X_{i,j}^{t}\\right| \\\\ \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003Ewhere \u003Cinline-formula\u003E\u003Ctex-math id=\"M40\"\u003E$$ t $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E denotes the number of iterations. After using this formula to improve the position update strategy of the algorithm, all CP individuals exchanged information with the optimal individuals in each exploration phase. Simultaneously, the golden section coefficient gradually reduced the search space of the CP individuals. By controlling the moving distance and direction of the CP individuals, the CPO algorithm was optimized, further coordinating the algorithm's global exploration and local exploitation abilities.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s2-3-3\" class=\"article-Section\"\u003E\u003Ch4 \u003E2.3.3. Adaptive weight strategy\u003C\u002Fh4\u003E\u003Cp class=\"\"\u003EWhen executing the third defense strategy, the search step of the CP individual was not set in the original algorithm, resulting in excessive freedom while running the algorithm. The adaptive weight strategy can dynamically adjust the optimal position\u003Csup\u003E[\u003Ca href=\"#b39\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b39\"\u003E39\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, thereby effectively enhancing the convergence effect and local exploitation ability of the CPO algorithm. This adjustment ensures that individuals with CP maintain a relatively safe distance from predators while executing the third defense strategy. Therefore, this study constructed an adaptive strategy that adjusted the weight coefficient \u003Cinline-formula\u003E\u003Ctex-math id=\"M41\"\u003E$$ \\omega $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E based on the iteration count, allowing CP individuals to utilize different weights for optimal search lengths at different stages. The \u003Cinline-formula\u003E\u003Ctex-math id=\"M42\"\u003E$$ \\omega $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is obtained as\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(10)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E10\"\u003E $$ \\begin{equation} \\omega=1-\\cosh\\left(\\left(\\exp(t\u002FT_{\\max})\\right)\u002F\\exp(1)-1\\right)^2 \\\\ \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003Ewhere \u003Cinline-formula\u003E\u003Ctex-math id=\"M43\"\u003E$$ \\text{cosh()} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E denotes the hyperbolic cosine function, and \u003Cinline-formula\u003E\u003Ctex-math id=\"M44\"\u003E$$ T_{\\max} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E denotes the maximum number of iterations.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s2-3-4\" class=\"article-Section\"\u003E\u003Ch4 \u003E2.3.4. Variable spiral search strategy\u003C\u002Fh4\u003E\u003Cp class=\"\"\u003EInspired by the whale optimization algorithm (WOA)\u003Csup\u003E[\u003Ca href=\"#b40\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b40\"\u003E40\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, the variable spiral search strategy adjusts the original spiral parameters to become variable parameters that change with each iteration. This adjustment allows the algorithm to perform extensive searches in the early phase and an elaborate exploration of a small area in the late stage\u003Csup\u003E[\u003Ca href=\"#b41\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b41\"\u003E41\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, enhancing its local exploitation ability in the fourth defense strategy. In this study, by constructing a variable spiral search strategy, CP individuals continued to search nearby after reaching the local optimal solution. This approach compensates for the unclear convergence effect of the original CPO during local exploration, which prevents deviations in the prediction accuracy of the model in the late stages of rail corrugation development. This strategy is established as\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(11)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E11\"\u003E $$ \\begin{equation} Z=X_{best}\\begin{pmatrix}t\\end{pmatrix}\\times\\begin{pmatrix}\\exp\\begin{pmatrix}zl\\end{pmatrix}\\times\\cos\\begin{pmatrix}2\\pi l\\end{pmatrix}\\end{pmatrix}+X_{best}\\begin{pmatrix}t\\end{pmatrix} \\\\ \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(12)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E12\"\u003E $$ \\begin{equation} z=\\exp\\bigl(k\\cos\\bigl(\\pi t\u002FT_{\\max}\\bigr)\\bigr) \\\\ \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003Ewhere \u003Cinline-formula\u003E\u003Ctex-math id=\"M45\"\u003E$$ X_{best} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E denotes the best fitness value, \u003Cinline-formula\u003E\u003Ctex-math id=\"M46\"\u003E$$ l $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E represents a random number between -1 and 1, and \u003Cinline-formula\u003E\u003Ctex-math id=\"M47\"\u003E$$ k $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E represents a variable parameter that should be set according to the specific strategy.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EBased on the above analysis, a flowchart of the MICPO algorithm is constructed [\u003Ca href=\"#Figure7\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure7\"\u003EFigure 7\u003C\u002Fa\u003E], where \u003Cinline-formula\u003E\u003Ctex-math id=\"M48\"\u003E$$ N $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M49\"\u003E$$ T_{max} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E represent the population size and maximum number of function evaluations, respectively. \u003Cinline-formula\u003E\u003Ctex-math id=\"M50\"\u003E$$ T_{f} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E indicates a constant between 0 and 1, \u003Cinline-formula\u003E\u003Ctex-math id=\"M51\"\u003E$$ t $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E denotes the number of current iterations, and \u003Cinline-formula\u003E\u003Ctex-math id=\"M52\"\u003E$$ i $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is the current \u003Cinline-formula\u003E\u003Ctex-math id=\"M53\"\u003E$$ i $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M54\"\u003E$$ th $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E individual. In the first iteration, all CP individuals passed through the position of the initialization solution and adopted a defense strategy to obtain the current optimal candidate solution. Subsequently, the algorithm entered the next iteration. First, the defense factor and the population number \u003Cinline-formula\u003E\u003Ctex-math id=\"M55\"\u003E$$ N $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E were updated. Then, the CP individuals continue to search for the best candidate solution of the model according to the selected defense strategy. This process was repeated until the iterations were complete. Consequently, the optimal solution, which represents the best parameter of the model, was obtained and substituted into the SA-BiTCN-BiGRU hybrid network to optimize the prediction performance of the model.\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure7\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure7\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU\" src=\"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-7.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 7. Flowchart of MICPO algorithm. MICPO: Multi-strategy improved crested porcupine optimizer.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s2-4\" class=\"article-Section\"\u003E\u003Ch3 \u003E2.4. Algorithm validation\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003EIn this study, six benchmark functions were used to conduct the optimization experiments. The MICPO algorithm was compared with the CPO\u003Csup\u003E[\u003Ca href=\"#b36\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b36\"\u003E36\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, WOA\u003Csup\u003E[\u003Ca href=\"#b40\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b40\"\u003E40\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, rime optimization algorithm (RIME)\u003Csup\u003E[\u003Ca href=\"#b42\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b42\"\u003E42\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, grey wolf optimizer (GWO)\u003Csup\u003E[\u003Ca href=\"#b43\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b43\"\u003E43\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, and dung-beetle optimizer (DBO)\u003Csup\u003E[\u003Ca href=\"#b44\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b44\"\u003E44\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E algorithm to observe their optimal fitness values and convergence speed within a specified number of iterations, and verify the improvement effect of MICPO on the original CPO. \u003Ca href=\"#Table1\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Table1\"\u003ETable 1\u003C\u002Fa\u003E provides a detailed definition of the benchmark functions. F1-F3 are single-peak functions used to evaluate the local search capability of the algorithm. F4 is a multipeak function with multiple local optimal values and requires a higher convergence performance of the algorithm. This function has important reference significance in the evaluation algorithm. F5 and F6 are the combined benchmark functions used to evaluate the global exploitation capacity of an algorithm. In this study, the population size of the experimental algorithm was set to 30, and each algorithm was optimized 100 times.\u003C\u002Fp\u003E\u003Cdiv id=\"Table1\" class=\"Figure-block\"\u003E\u003Cdiv class=\"table-note\"\u003E\u003Cspan class=\"\"\u003ETable 1\u003C\u002Fspan\u003E\u003Cp class=\"\"\u003EDetailed information on benchmark function\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"table-responsive article-table\"\u003E\u003Ctable class=\"a-table\"\u003E\u003Cthead\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EID\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EBenchmark function\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EDomain and dimensions\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EOptimal value\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003C\u002Fthead\u003E\u003Ctbody\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003EF1\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M56\"\u003E$$ f_1(x)=\\sum\\limits_{i=1}^{n}{x_{i}^{2}} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M57\"\u003E$$ \\begin{bmatrix}-100, 100\\end{bmatrix}^{30} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E0\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EF2\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M58\"\u003E$$ f_1(x) = \\sum\\limits_{i = 1}^n {x_i^2} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M59\"\u003E$$ \\begin{bmatrix}-100, 100\\end{bmatrix}^{30} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EF3\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M60\"\u003E$$ f_3(x) = \\sum\\limits_{i = 1}^n {ix_i^4 + \\text{random}[0,1)} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M61\"\u003E$$ \\begin{bmatrix}-1.28, 1.28\\end{bmatrix}^{30} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EF4\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M62\"\u003E$$ f_4(x) = - 20\\exp \\left( { - 0.2\\sqrt {\\frac{1}{n}\\sum\\limits_{i = 1}^n {x_i^2} } } \\right) - \\exp \\left( {\\frac{1}{n}\\sum\\limits_{i = 1}^n {\\cos } (2\\pi {x_i})} \\right) + 20 + e $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M63\"\u003E$$ \\begin{bmatrix}-32, 32\\end{bmatrix}^{30} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EF5\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M64\"\u003E$$ f_5(x) = \\sum\\limits_{i = 1}^n {\\left[ {{a_i} - \\frac{{{x_1}(b_i^2 + {b_1}{x_2})}}{{b_i^2 + {b_1}{x_3} + {x_4}}}} \\right]} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M65\"\u003E$$ \\begin{bmatrix}-5, 5\\end{bmatrix}^{4} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M66\"\u003E$$ 3.075\\times10^{-4} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003EF6\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M67\"\u003E$$ f_6(x) = - \\sum\\limits_{i = 1}^{10} {{{\\left[ {(x - {a_i}){{(x - {a_i})}^T} + {c_i}} \\right]}^{ - 1}}} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M68\"\u003E$$ \\begin{bmatrix}0, 10\\end{bmatrix}^{4} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E-10\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003C\u002Ftbody\u003E\u003C\u002Ftable\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"table_footer\"\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003EAs shown in \u003Ca href=\"#Figure8\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure8\"\u003EFigure 8\u003C\u002Fa\u003E, the convergence performance of the MICPO algorithm is effectively proven.\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure8\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure8\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU\" src=\"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-8.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 8. Convergence curves of different algorithms under different benchmark functions. (A-F) correspond to benchmark functions F1-F6, respectively.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003EFrom the convergence curve presented in \u003Ca href=\"#Figure8\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure8\"\u003EFigure 8\u003C\u002Fa\u003E, the CPO algorithm exhibits poor convergence performance and easily falls into the local optima, indicating that improvements in the CPO algorithm are necessary. In the unimodal function test shown in \u003Ca href=\"#Figure8\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure8\"\u003EFigure 8A\u003C\u002Fa\u003E-\u003Ca href=\"#Figure8\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure8\"\u003EC\u003C\u002Fa\u003E, the RIME, WOA, and the other optimization algorithms fell into local optima and slowly converged, indicating that the MICPO algorithm has certain competitive advantages over other optimization algorithms in solving unimodal high-dimensional functions. In the multipeak test function F4, the MICPO algorithm [\u003Ca href=\"#Figure8\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure8\"\u003EFigure 8D\u003C\u002Fa\u003E], demonstrates an advantage by being the closest to the optimal solution within the specified number of iterations, which validates its effectiveness in improving the CPO algorithm, as well as its superiority in search accuracy and convergence speed. In the combined function test shown in \u003Ca href=\"#Figure8\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure8\"\u003EFigure 8E\u003C\u002Fa\u003E and \u003Ca href=\"#Figure8\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure8\"\u003EF\u003C\u002Fa\u003E, the MICPO and CPO algorithms demonstrate superior convergence performance compared to the RIME algorithm and other optimization algorithms, indicating their advancement in global optimization.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s3\" class=\"article-Section\"\u003E\u003Ch2 \u003E3. RESULTS\u003C\u002Fh2\u003E\u003Cdiv id=\"s3-1\" class=\"article-Section\"\u003E\u003Ch3 \u003E3.1. Experimental setup and rail corrugation dataset preprocessing\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003EFirst, the code was written and debugged on a PyCharm platform, and the running environment consisted of a processor (Intel i7-12700H), 16 GB of random-access memory (RAM), a graphics card (RTX 3060), and a software environment with TensorFlow 2.13.0 and Python 3.9.18. The experimental data in this study were actual measurement data from a railway section in China. A track inspection car was used to collect vibration signals from a typical steel rail segment with corrugation, covering damage from slight to severe stages. These signals demonstrate the progression of corrugation damage\u003Csup\u003E[\u003Ca href=\"#b21\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b21\"\u003E21\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. We collected 98 vibration samples on-site at the same time interval throughout the entire lifecycle of the rail after several months of continuous periodic testing, with each vibration sample containing 3, 000 sample points; therefore, the original sample contained 98 × 3, 000 data points. These data points represent the initial corrugation on the rail surface to the rail scrap. The overall vibration amplitude gradually increased with collection times, indicating that the deterioration degree of corrugation damage was worsening, reflecting the evolution of rail corrugation damage from budding to deterioration.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EFirst, each collected sample was subjected to multidomain feature extraction to obtain 26 feature indicators that reflected the evolution of corrugation. The dimensions of the samples were 98 × 26. Subsequently, the \u003Cinline-formula\u003E\u003Ctex-math id=\"M69\"\u003E$$ M $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, \u003Cinline-formula\u003E\u003Ctex-math id=\"M70\"\u003E$$ S $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, and \u003Cinline-formula\u003E\u003Ctex-math id=\"M71\"\u003E$$ R $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E of each feature index were calculated; \u003Cinline-formula\u003E\u003Ctex-math id=\"M72\"\u003E$$ C $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E was used to adaptively screen out the eight features with higher scores, and the corrugation optimal feature subset with a sample dimension of 98 × 8 was obtained. PCA was used to fuse the optimal feature subset, resulting in a two-dimensional principal component vector with a total contribution rate of 97%. The sample dimensions of the corrugation data were 98 × 2. Finally, the MD was used to calculate the difference between the first column of the sample data and the subsequent 97 columns of sample data, resulting in 98 × 1 one-dimensional data. The corrugation CHI was obtained after smoothing to minimize the negative impact of outliers on the prediction of the evolution trend of rail corrugation.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s3-2\" class=\"article-Section\"\u003E\u003Ch3 \u003E3.2. Validation of the CHI construction method\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003ETo demonstrate the effectiveness and advantages of the method proposed for constructing the corrugation CHI, several commonly used methods for constructing health indicators were selected for comparison, including the RMS, PCA, and locally linear embedding (LLE) fusion indicators. Two fusion indicators were constructed using the optimal feature subset described in Section 3.1. The rail corrugation health indicator constructed using these four methods after smoothing is shown in \u003Ca href=\"#Figure9\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure9\"\u003EFigure 9\u003C\u002Fa\u003E.\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure9\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure9\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU\" src=\"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-9.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 9. Health indicators of rail corrugation constructed by different methods. (A-D) correspond to RMS, PCA, LLE and CHI methods, respectively. RMS: Root mean square; PCA: principal component analysis; LLE: locally linear embedding; CHI: comprehensive health indicator.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003E\u003Ca href=\"#Figure9\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure9\"\u003EFigure 9\u003C\u002Fa\u003E shows that these indicators are relatively sensitive to changes in the initial corrugation damage. However, the RMS indicator exhibits a larger overall fluctuation range, with the index value declining in the later stages of the corrugation evolution and deviating from the actual situation. The amplitude of the PCA indicator fluctuates significantly between the middle and late stages. The LLE indicator oscillates excessively in the early stages and becomes more stable in the middle and late stages, which is different from the actual situation. However, the CHI constructed in this study showed a better overall trend with fewer fluctuations. The indicator shows a sudden increase when the corrugation damage approached a qualitative change in the later stage, which aligns with the actual evolution law of on-site rail corrugation damage. The corrugation health indicator constructed by CHI is more consistent with the changing trend of the real-world data on rail corrugation vibration signals.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EFurthermore, the \u003Cinline-formula\u003E\u003Ctex-math id=\"M73\"\u003E$$ M $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, \u003Cinline-formula\u003E\u003Ctex-math id=\"M74\"\u003E$$ S $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, \u003Cinline-formula\u003E\u003Ctex-math id=\"M75\"\u003E$$ R $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, and \u003Cinline-formula\u003E\u003Ctex-math id=\"M76\"\u003E$$ C $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E [established by Equations (1)-(4)] were used to evaluate the HIs constructed using the four different methods. The results are listed in \u003Ca href=\"#Table2\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Table2\"\u003ETable 2\u003C\u002Fa\u003E.\u003C\u002Fp\u003E\u003Cdiv id=\"Table2\" class=\"Figure-block\"\u003E\u003Cdiv class=\"table-note\"\u003E\u003Cspan class=\"\"\u003ETable 2\u003C\u002Fspan\u003E\u003Cp class=\"\"\u003EEvaluation results of health indicators\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"table-responsive article-table\"\u003E\u003Ctable class=\"a-table\"\u003E\u003Cthead\u003E\u003Ctr\u003E\u003Ctd align=\"center\" style=\"class:table_top_border\"\u003E\u003Cb\u003EIndicator\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd align=\"center\" style=\"class:table_top_border\"\u003E\u003Cb\u003E\u003Ci\u003EM\u003C\u002Fi\u003E\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd align=\"center\" style=\"class:table_top_border\"\u003E\u003Cb\u003E\u003Ci\u003ES\u003C\u002Fi\u003E\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd align=\"center\" style=\"class:table_top_border\"\u003E\u003Cb\u003E\u003Ci\u003ER\u003C\u002Fi\u003E\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd align=\"center\" style=\"class:table_top_border\"\u003E\u003Cb\u003E\u003Ci\u003EC\u003C\u002Fi\u003E\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003C\u002Fthead\u003E\u003Ctbody\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003ERMS\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E0.1134\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E0.8336\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E0.9043\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E0.6171\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EPCA\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.1546\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.8706\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.9021\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.6424\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ELLE\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.1753\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.8983\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.9098\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.6611\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003EProposed indicator\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E0.2165\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E0.9322\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E0.928\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E0.6922\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003C\u002Ftbody\u003E\u003C\u002Ftable\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"table_footer\"\u003E\u003Cdiv class=\"table_footer_note\"\u003E\u003Cp class=\"para\"\u003ERMS: Root mean square; PCA: principal component analysis; LLE: linear embedding.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003EThrough the comparison of various indicators in \u003Ca href=\"#Table2\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Table2\"\u003ETable 2\u003C\u002Fa\u003E, the constructed CHI achieved optimal performance in all cases, with the highest comprehensive evaluation function \u003Cinline-formula\u003E\u003Ctex-math id=\"M77\"\u003E$$ C $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. Therefore, this indicator is considered suitable for reflecting the evolution trend of rail corrugation.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s3-3\" class=\"article-Section\"\u003E\u003Ch3 \u003E3.3. Performance evaluation indicators\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003EThe root mean square error (RMSE), mean square error (MSE), and mean absolute error (MAE) were used to evaluate the performance of the model. These indices reflect the prediction effect by calculating the error between the predicted and true CHI values. Simultaneously, to address inconsistencies among the different indicator dimensions, \u003Cinline-formula\u003E\u003Ctex-math id=\"M78\"\u003E$$ R^{2} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E was added as an evaluation criterion. The indices are estimated using\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(13)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E13\"\u003E $$ \\begin{equation} {RMSE = \\sqrt {\\frac{1}{N}\\mathop \\sum \\nolimits_{i = 1}^N {{({x_i} - {y_i})}^2}} } \\\\ \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(14)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E14\"\u003E $$ \\begin{equation} MSE=\\frac{\\sum_{\\mathrm{i}=1}^{\\mathrm{N}}\\left(x_i-y_i\\right)^2}N \\\\ \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(15)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E15\"\u003E $$ \\begin{equation} MAE=\\frac{\\sum_{i=1}^N\\left|x_i-y_i\\right|}{N} \\\\ \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(16)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E16\"\u003E $$ \\begin{equation} R^{2}=1-\\frac{\\sum_{i=1}^{N}(x_{i}-y_{i})^{2}}{\\sum_{i=1}^{N}(x_{i}-\\overline{x}_{i})^{2}} \\\\ \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003Ewhere \u003Cinline-formula\u003E\u003Ctex-math id=\"M79\"\u003E$$ {x_i} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E represents the true corrugation CHI value, \u003Cinline-formula\u003E\u003Ctex-math id=\"M80\"\u003E$$ {y_i} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E represents the predicted value by the model, \u003Cinline-formula\u003E\u003Ctex-math id=\"M81\"\u003E$$ {N} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is the data length of the corrugation CHI, and \u003Cinline-formula\u003E\u003Ctex-math id=\"M82\"\u003E$$ \\bar{x}_i $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E denotes the average of the true value.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s3-4\" class=\"article-Section\"\u003E\u003Ch3 \u003E3.4. Predictive experimental analysis of rail corrugation\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003EAfter obtaining the CHI of the corrugation damage according to Section 3.1, the corrugation CHI data with a length of 98 can be expressed as \u003Cinline-formula\u003E\u003Ctex-math id=\"M83\"\u003E$$ \\{x_1,x_2,\\cdots,x_{98}\\} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. Our study used 75% of the data as the training set and the remainder as the test set; thus, the training set was \u003Cinline-formula\u003E\u003Ctex-math id=\"M84\"\u003E$$ \\{x_1,x_2,\\cdots,x_{n}\\} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and the test set was \u003Cinline-formula\u003E\u003Ctex-math id=\"M85\"\u003E$$ \\{x_{n+1},x_{n+2},\\cdots,x_{98}\\} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. The SA-BiTCN-BiGRU model was trained using the training set, whereas the test set was used to verify the effect of the model in predicting the evolution trend of rail corrugation. Subsequently, according to the input step length \u003Cinline-formula\u003E\u003Ctex-math id=\"M86\"\u003E$$ t $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E of the prediction model, the single-step sliding window approach was used for forecasting, using the corrugation initial data from the CHI to predict the subsequent evolution of the corrugation. For example, the input of the first sample was \u003Cinline-formula\u003E\u003Ctex-math id=\"M87\"\u003E$$ \\begin{Bmatrix}x_1,x_2,\\cdots,x_t\\end{Bmatrix} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, yielding the prediction result of \u003Cinline-formula\u003E\u003Ctex-math id=\"M88\"\u003E$$ y_{t+1} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. Then, the prediction was gradually conducted to obtain the prediction result \u003Cinline-formula\u003E\u003Ctex-math id=\"M89\"\u003E$$ y_{98} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E of the last sample. The error between the predicted value of the model and the input CHI was calculated to evaluate the prediction performance of the model. Other variables that may have affected the experimental results were controlled to ensure that the observed changes were caused by the proposed method. \u003Ca href=\"#Table3\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Table3\"\u003ETable 3\u003C\u002Fa\u003E presents the network model parameters used for predicting the evolution trend of the rail corrugation.\u003C\u002Fp\u003E\u003Cdiv id=\"Table3\" class=\"Figure-block\"\u003E\u003Cdiv class=\"table-note\"\u003E\u003Cspan class=\"\"\u003ETable 3\u003C\u002Fspan\u003E\u003Cp class=\"\"\u003EMain parameter settings of proposed network\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"table-responsive article-table\"\u003E\u003Ctable class=\"a-table\"\u003E\u003Cthead\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EParameter\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EValue\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003C\u002Fthead\u003E\u003Ctbody\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003EEpochs\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E1000\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EBatch_size\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E128\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EOptimizer\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003EAdam\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ELeaky rate\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.01\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ELearning rate\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M90\"\u003E$$ [1\\times10^{-4},1\\times10^{-2}] $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EDropout rate\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M91\"\u003E$$ [1\\times10^{-3},1\\times10^{-2}] $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EKernel size\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E[2, 7]\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ENumber of filters\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E[8, 128]\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003ENumber of BiGRU hidden unit\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E[8, 128]\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003C\u002Ftbody\u003E\u003C\u002Ftable\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"table_footer\"\u003E\u003Cdiv class=\"table_footer_note\"\u003E\u003Cp class=\"para\"\u003EBiGRU: Bidirectional gated recurrent unit.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s3-4-1\" class=\"article-Section\"\u003E\u003Ch4 \u003E3.4.1. Ablation experiment\u003C\u002Fh4\u003E\u003Cp class=\"\"\u003EA comprehensive quantitative analysis of the structure and function of the proposed network was conducted to highlight the effects of each module on the MICPO-SA-BiTCN-BiGRU network. The results are summarized in \u003Ca href=\"#Table4\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Table4\"\u003ETable 4\u003C\u002Fa\u003E.\u003C\u002Fp\u003E\u003Cdiv id=\"Table4\" class=\"Figure-block\"\u003E\u003Cdiv class=\"table-note\"\u003E\u003Cspan class=\"\"\u003ETable 4\u003C\u002Fspan\u003E\u003Cp class=\"\"\u003EAblation experiment prediction errors\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"table-responsive article-table\"\u003E\u003Ctable class=\"a-table\"\u003E\u003Cthead\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EPrediction model\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003ERMSE\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EMSE\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EMAE\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003E\u003Ci\u003ER\u003C\u002Fi\u003E\u003Csup\u003E2\u003C\u002Fsup\u003E\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003C\u002Fthead\u003E\u003Ctbody\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003ETCN\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E0.415\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E0.172\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E0.309\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E0.82\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ETCN-GRU\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.341\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.117\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.227\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.878\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EBiTCN-BiGRU\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.315\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.099\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.194\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.896\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ESA-BiTCN-BiGRU\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.239\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.057\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.13\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.94\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ECPO-SA-BiTCN-BiGRU\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.171\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.029\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.119\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.969\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003EProposed model\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E0.119\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E0.014\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E0.095\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E0.985\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003C\u002Ftbody\u003E\u003C\u002Ftable\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"table_footer\"\u003E\u003Cdiv class=\"table_footer_note\"\u003E\u003Cp class=\"para\"\u003ERMSE: Root mean square error; MSE: mean square error; MAE: mean absolute error; TCN: temporal convolutional network; GRU: gated recurrent unit; BiTCN: bidirectional temporal convolutional network; BiGRU: bidirectional gated recurrent unit; SA: self-attention; CPO: crested porcupine optimizer.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003EFrom \u003Ca href=\"#Table4\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Table4\"\u003ETable 4\u003C\u002Fa\u003E, the RMSE, MSE, and MAE decreased by 17.8%, 32%, and 26.5%, respectively, from TCN to TCN-GRU, whereas \u003Cinline-formula\u003E\u003Ctex-math id=\"M92\"\u003E$$ R^{2} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E increased by 7.1%. If a bidirectional network structure (BiTCN-BiGRU model) was added, the RMSE, MSE and MAE further decreased by 7.6%, 15.4%, and 14.5%, respectively, whereas \u003Cinline-formula\u003E\u003Ctex-math id=\"M93\"\u003E$$ R^{2} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E increased by 2.1%. This indicates that the structure further improved its prediction by considering the information on the forward and backward evolution of rail corrugation. When SA was introduced into the BiTCN-BiGRU model, the RMSE, MSE, and MAE decreased by 24.1%, 42.4%, and 33%, respectively, and \u003Cinline-formula\u003E\u003Ctex-math id=\"M94\"\u003E$$ R^{2} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E increased by 4.91%. This indicates that the introduction of SA improved the feature expression capability of the network and reduced its dependence on irrelevant information. To reduce the impact of the artificial selection of network hyperparameters on the prediction results, the CPO algorithm was added for model optimization. Consequently, the RMSE, MSE, and MAE decreased by 28.5%, 49.1%, and 8.5%, respectively, and \u003Cinline-formula\u003E\u003Ctex-math id=\"M95\"\u003E$$ R^{2} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E increased by 3.1%. Subsequently, the search strategy of the original CPO algorithm was improved to effectively alleviate the problem of local convergence. Consequently, the RMSE, MSE, and MAE decreased by 30.4%, 51.7%, and 20.2%, respectively, and \u003Cinline-formula\u003E\u003Ctex-math id=\"M96\"\u003E$$ R^{2} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E increased by 1.7%, indicating that the optimization algorithm and its improved strategy were effective for model prediction.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EThe visualization results of the model ablation experiment presented in \u003Ca href=\"#Figure10\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure10\"\u003EFigure 10\u003C\u002Fa\u003E show that the proposed method (brown line) closely matches the true value (green line), particularly during the model testing phase, thus effectively predicting the evolution process of corrugation in the later stages of development. Additionally, the proposed method shows a higher local prediction accuracy compared to the other models.\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure10\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure10\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU\" src=\"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-10.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 10. Model ablation experiment: prediction results of evolution trend of rail corrugation.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s3-4-2\" class=\"article-Section\"\u003E\u003Ch4 \u003E3.4.2. Comparison experiment\u003C\u002Fh4\u003E\u003Cp class=\"\"\u003ETo verify the timeliness of the MICPO-SA-BiTCN-BiGRU network model in predicting the evolution trend of rail corrugation, we used the network models from recently published studies to predict the evolution trend of rail corrugation and quantitatively analyze and compare the predicted results with those of the proposed model. The results are listed in \u003Ca href=\"#Table5\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Table5\"\u003ETable 5\u003C\u002Fa\u003E.\u003C\u002Fp\u003E\u003Cdiv id=\"Table5\" class=\"Figure-block\"\u003E\u003Cdiv class=\"table-note\"\u003E\u003Cspan class=\"\"\u003ETable 5\u003C\u002Fspan\u003E\u003Cp class=\"\"\u003EComparison experiment prediction errors\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"table-responsive article-table\"\u003E\u003Ctable class=\"a-table\"\u003E\u003Cthead\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EPrediction model\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003ERMSE\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EMSE\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EMAE\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003E\u003Ci\u003ER\u003C\u002Fi\u003E\u003Csup\u003E2\u003C\u002Fsup\u003E\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003C\u002Fthead\u003E\u003Ctbody\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003ETCN-GRU-attention\u003Csup\u003E[\u003Ca href=\"#b20\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b20\"\u003E20\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E0.351\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E0.123\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E0.223\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E0.872\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ECNN-BiGRU-attention\u003Csup\u003E[\u003Ca href=\"#b23\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b23\"\u003E23\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.36\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.13\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.276\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.864\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ECNN-GRU\u003Csup\u003E[\u003Ca href=\"#b45\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b45\"\u003E45\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.448\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.201\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.351\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.79\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ECNN-LSTM-attention\u003Csup\u003E[\u003Ca href=\"#b46\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b46\"\u003E46\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.377\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.142\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.29\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.852\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ESA-TCN-LSTM\u003Csup\u003E[\u003Ca href=\"#b47\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b47\"\u003E47\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.295\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.087\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.196\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.909\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003EProposed model\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E0.119\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E0.014\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E0.095\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E0.985\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003C\u002Ftbody\u003E\u003C\u002Ftable\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"table_footer\"\u003E\u003Cdiv class=\"table_footer_note\"\u003E\u003Cp class=\"para\"\u003ERMSE: Root mean square error; MSE: mean square error; MAE: mean absolute error; TCN: temporal convolutional network; GRU: gated recurrent unit; CNN: convolutional neural network; BiGRU: bidirectional gated recurrent unit; LSTM: long short-term memory; SA: self-attention.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003EFrom \u003Ca href=\"#Table5\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Table5\"\u003ETable 5\u003C\u002Fa\u003E, the proposed model has a lower prediction error than the other models. Consequently, the RMSE decreased by 66.1% to 73.4%, the MSE by 88.6% to 93%, and the MAE by 57.4% to 72.9%. Conversely, the \u003Cinline-formula\u003E\u003Ctex-math id=\"M97\"\u003E$$ R^{2} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E increased by 8.36% to 24.7%. This indicates that the MICPO-SA-BiTCN-BiGRU model has a suitable architecture and accurately predicts the evolutionary trend of corrugation.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003ETo discover the evolutionary trend of corrugation more intuitively, a visualization from the comparative experiment is shown in \u003Ca href=\"#Figure11\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure11\"\u003EFigure 11\u003C\u002Fa\u003E.\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure11\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure11\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU\" src=\"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-11.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 11. Model comparison experiment: prediction results of evolution trend of rail corrugation.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003E\u003Ca href=\"#Figure11\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure11\"\u003EFigure 11\u003C\u002Fa\u003E shows that the development of corrugation damage on the measured road section exists in relatively evident stages. Therefore, we divided the data collected from measurements 1 to 40 into the early stage of rail corrugation evolution, during which the CHI value increased by approximately 3.4, indicating rapid development. The data from measurements 41 to 85 were categorized as the middle stage of rail corrugation development. During this period, the CHI value increased by approximately 0.6, with the development of corrugation leveling off and fluctuations rising slowly. This indicates that the damage caused by corrugation to the rail began to intensify, and the rail was approaching a critical state. The data from measurements 86 to 98, categorized as the late stage of corrugation development, showed that the CHI value increased by approximately 1.6. During this period, the degree of corrugation damage deterioration showed a sudden increase, indicating a sharp decline in the health of the rail within a short period, thus necessitating prompt measures to curb its development.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EOverall, the prediction trends of the models were similar; however, the proposed model was the most accurate for local prediction. In particular, during the early and late phases of corrugation damage, the model effectively captured the evolution trend of rail corrugation damage, with its predicted value closely aligned with the real value.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s4\" class=\"article-Section\"\u003E\u003Ch2 \u003E4. DISCUSSION\u003C\u002Fh2\u003E\u003Cp class=\"\"\u003EPredicting the evolutionary trend of rail corrugation is critical for the safe operation and maintenance of railways. To address the difficulties involved in accurately evaluating the evolution state of corrugation, a method was proposed to predict the evolution trend of corrugation. By analyzing the existing on-site data on rail corrugation, the CHI and SA-BiTCN-BiGRU hybrid network models were constructed to predict the evolution process of corrugation in the time dimension. The results were better than those of existing studies.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EHowever, constructing the corrugated CHI partly relies on manual experience, which is highly subjective and results in limited accuracy and standardization. In future studies, we will attempt to combine multi-source data such as on-site rail corrugation images, vibrations, and profile data to predict the evolution trend of rail corrugation. The proposed method improves the generality and reliability of our study by combining more comprehensive corrugation damage information, ensuring the safe operation of the corresponding railway line.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EFurther, we recognize the importance of predicting the location and duration of rail corrugations. Yet, the proposed method was not effective in predicting the location of rail corrugation, and the collected dataset made the prediction of duration challenging. In fault prediction and health management, most existing research focuses on predicting the development and evolution of rail corrugation, with significantly few studies addressing its location. Nevertheless, numerous scholars have studied the detection of rail corrugation positions. For instance, Yang \u003Ci\u003Eet al\u003C\u002Fi\u003E. proposed an intelligent real-time detection method for rail corrugation using machine vision and CNN; Li \u003Ci\u003Eet al\u003C\u002Fi\u003E. proposed an intelligent detection method for rail corrugation using signal decomposition and the entropy theory\u003Csup\u003E[\u003Ca href=\"#b48\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b48\"\u003E48\u003C\u002Fa\u003E,\u003Ca href=\"#b49\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b49\"\u003E49\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. In our future work, we will aim to combine spatial data to predict the location of rail corrugation and detect rail damage promptly. Additionally, we collected annotated data on the timing and duration of rail corrugation, which can assist in predicting its duration. These efforts will significantly improve the depth of our research and represent an important direction for future studies.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s5\" class=\"article-Section\"\u003E\u003Ch2 \u003E5. CONCLUSIONS\u003C\u002Fh2\u003E\u003Cp class=\"\"\u003EIn this study, we proposed an intelligent prediction method for the evolutionary trend of rail corrugation based on SA, BiTCN, and BiGRU.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EFirst, a health indicator reflecting the evolution state of the corrugation was obtained using the defined method for constructing corrugation CHI. The experimental results validated the effectiveness of the CHI. Second, we effectively demonstrated the interpretability and predictive ability of the proposed bidirectional hybrid network, SA-BiTCN-BiGRU, through an ablation experiment. Third, by using the MICPO algorithm, the optimal values of the key hyperparameters of the SA-BiTCN-BiGRU model were determined, thereby improving the prediction accuracy of the corrugation evolution trend. The findings demonstrated the high convergence capabilities of the MICPO algorithm compared to other swarm intelligent optimization algorithms. The ablation experiment strongly verified the positive role of the MICPO algorithm in improving model prediction results. Finally, the results of the model comparison confirmed that the MICPO-SA-BiTCN-BiGRU model is efficient. The proposed method is significant for railway maintenance, as it effectively predicts the future development trend of rail corrugation and provides a scientific basis for railway maintenance decisions.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s6\" class=\"article-Section\"\u003E\u003Ch2 \u003EDECLARATIONS\u003C\u002Fh2\u003E\u003Cdiv id=\"s6-1\" class=\"article-Section\"\u003E\u003Ch3 \u003EAcknowledgments\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003EWe thank the Editor-in-Chief and all reviewers for their comments.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s6-2\" class=\"article-Section\"\u003E\u003Ch3 \u003EAuthors' contributions\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003EConducted experimental analysis and manuscript writing: Yang WH\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EGuided on the overall framework and implementation steps of this research, and proposed a train of thought for the general research objectives: Liu JH, Zhang CF\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EGuided English writing: He J\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EProvided technical support: Wang ZM, Jia L\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EProvided dataset support: Yang WW\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s6-3\" class=\"article-Section\"\u003E\u003Ch3 \u003EAvailability of data and materials\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003EThe data are available upon request. If needed, please contact the corresponding author by email.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s6-4\" class=\"article-Section\"\u003E\u003Ch3 \u003EFinancial support and sponsorship\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003EThis research was funded by the National Key Research and Development Program (Grant No. 2021YFF0501101), the National Natural Science Foundation of China (Grant Nos. 52272347, 62303178), Key Scientific Research Project of the Hunan Provincial Department of Education (Grant No. 22A0391), the Natural Science Foundation of the Hunan Province (Grant No. 2024JJ7132).\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s6-5\" class=\"article-Section\"\u003E\u003Ch3 \u003EConflicts of interest\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003EYang WW is affiliated with Zhuzhou Qingyun Electric Locomotive Accessories Factory Co., Ltd., while the other authors have declared that they have no conflicts of interest.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s6-6\" class=\"article-Section\"\u003E\u003Ch3 \u003EEthical approval and consent to participate\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003ENot applicable.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s6-7\" class=\"article-Section\"\u003E\u003Ch3 \u003EConsent for publication\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003ENot applicable.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"s6-8\" class=\"article-Section\"\u003E\u003Ch3 \u003ECopyright\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003E© The Author(s) 2024.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E",translate:[{language:"en",new_title:bF,new_abstract:bG,new_keywords:"Mahalanobis distance, rail corrugation, evolution trend prediction, improved crested porcupine optimizer, hybrid time series network",is_check:j},{language:"cn",new_title:"基于自注意力双向TCN和GRU的铁路波痕演变趋势智能预测",new_abstract:"使用信号处理和深度学习对铁路脉动的演变趋势进行分析对铁路安全至关重要，因为目前传统方法难以捕捉脉动的复杂演变。本研究解决了准确捕捉这一趋势的挑战，这在很大程度上依赖于专家判断，提出了基于自注意（SA）、双向时间卷积网络（TCN）和双向门控循环单元（GRU）的智能预测方法。首先，使用多域特征提取和自适应特征筛选来获取最佳特征集合。然后，将这些特征与主成分分析（PCA）和马氏距离（MD）方法相结合，构建了反映铁路脉动演变的全面健康指标（CHI）。采用双向融合模型架构来捕捉脉动演变过程中正向和反向信息之间的时间相关性，同时在模型中嵌入SA来增强对关键信息的关注。结果是一个将双向TCN、双向GRU和SA结合的铁路脉动趋势预测网络。随后，构建了一个多策略改进的冠刺豪猪优化器（CPO）算法，用于自动获取最佳网络超参数。所提出的方法经过现场铁路脉动数据验证，表现出比其他先进方法更优越的预测性能。总之，所提出的方法能够准确预测铁路脉动的演变趋势，为现场铁路维护提供了有价值的工具。",new_keywords:"马哈拉诺比斯距离，轨道波痕，演化趋势预测，改进的冠毛豪猪优化器，混合时间序列网络",is_check:h},{language:"de",new_title:"Intelligente Vorhersage der Entwicklungstendenz von Schienenverwerfungen basierend auf selbst-aufmerksamkeitsbasiertem bidirektionalem TCN und GRU",new_abstract:"Die Analyse des Evolutionstrends von Schienenschwingungen mithilfe von Signalverarbeitung und Deep Learning ist entscheidend für die Sicherheit im Schienenverkehr, da aktuelle traditionelle Methoden Schwierigkeiten haben, die komplexe Evolution von Schwingungen zu erfassen. Diese Studie befasst sich mit der Herausforderung, diesen Trend genau zu erfassen, der maßgeblich vom Expertenurteil abhängt, indem sie eine intelligente Vorhersagemethode auf Basis von Self-Attention (SA), einem bidirektionalen zeitlichen Faltungsnetzwerk (TCN) und einer bidirektionalen Gated Recurrent Unit (GRU) vorschlägt. Zunächst wurden multidomänen Feature-Extraktion und adaptive Feature-Auswahl verwendet, um den optimalen Merkmalsatz zu erhalten. Diese Merkmale wurden dann mit der Hauptkomponentenanalyse (PCA) und der Mahalanobis-Distanzmethode kombiniert, um einen umfassenden Gesundheitsindikator (CHI) zu konstruieren, der die Evolution von Schienenschwingungen widerspiegelt. Eine bidirektionale Fusionsmodellarchitektur wurde verwendet, um die zeitlichen Korrelationen zwischen Vorwärts- und Rückwärtsinformationen während der Schwingungsevolution zu erfassen, wobei SA in das Modell eingebettet wurde, um den Fokus auf Schlüsselinformationen zu verstärken. Das Ergebnis war ein Netzwerk zur Vorhersage des Schienenschwingungstrends, das ein bidirektionales TCN, eine bidirektionale GRU und SA kombinierte. Anschließend wurde ein Multi-Strategie verbesserter Stachelschweinoptimierungsalgorithmus (CPO) konstruiert, um automatisch die optimalen Netzwerkhyperparameter zu erhalten. Die vorgeschlagene Methode wurde mit vor Ort vorhandenen Daten zu Schienenschwingungen validiert und zeigte eine überlegene Vorhersageleistung im Vergleich zu anderen fortschrittlichen Methoden. Zusammenfassend kann die vorgeschlagene Methode den Evolutionstrend von Schienenschwingungen genau vorhersagen und stellt ein wertvolles Werkzeug für die Wartung im Schienenbereich dar.",new_keywords:"Mahalanobis-Abstand, Schienenrippung, Evolutionstrendvorhersage, verbesserter Rattentriecher-Optimierer, hybrides Zeitreihennetzwerk",is_check:h},{language:"fa",new_title:"Prédiction intelligente de la tendance d'évolution de la corrélation des rails basée sur TCN bidirectionnel à auto-attention et GRU",new_abstract:"L'analyse de la tendance de l'évolution de la corrugation des rails en utilisant le traitement du signal et l'apprentissage en profondeur est cruciale pour la sécurité ferroviaire, car les méthodes traditionnelles actuelles peinent à capturer l'évolution complexe de la corrugation. Cette étude aborde le défi de capturer de manière précise cette tendance, qui repose fortement sur le jugement d'experts, en proposant une méthode de prédiction intelligente basée sur l'auto-attention (SA), un réseau de convolution temporelle bidirectionnelle (TCN) et une unité récurrente bidirectionnelle à porte (GRU). Tout d'abord, l'extraction de caractéristiques multidomaines et le criblage adaptatif des caractéristiques ont été utilisés pour obtenir l'ensemble optimal de caractéristiques. Ces caractéristiques ont ensuite été combinées avec l'analyse en composantes principales (ACP) et la distance de Mahalanobis (MD) pour construire un indicateur global de santé (CHI) qui reflète l'évolution de la corrugation des rails. Une architecture de modèle de fusion bidirectionnelle a été utilisée pour capturer les corrélations temporelles entre les informations avant et arrière pendant l'évolution de la corrugation, l'auto-attention étant intégrée dans le modèle pour renforcer la concentration sur les informations clés. Le résultat a été un réseau de prédiction de tendance de la corrugation des rails combinant un TCN bidirectionnel, un GRU bidirectionnel et l'auto-attention. Ensuite, un algorithme d'optimisation du porc-épic crêté amélioré à multiples stratégies (CPO) a été construit pour obtenir automatiquement les hyperparamètres optimaux du réseau. La méthode proposée a été validée avec des données de corrugation des rails sur site, démontrant des performances prédictives supérieures par rapport à d'autres méthodes avancées. En résumé, la méthode proposée peut prédire avec précision la tendance de l'évolution de la corrugation des rails, offrant un outil précieux pour la maintenance ferroviaire sur site.",new_keywords:"Distance de Mahalanobis, corrugation des rails, prédiction de tendance d'évolution, optimiseur amélioré du porc-épic crêté, réseau hybride de séries temporelles",is_check:h},{language:"jp",new_title:"自己注意双方向TCN和GRUを基にしたレールの波打ち進化傾向のインテリジェントな予測",new_abstract:"鉄道の安全性において、信号処理と深層学習を用いてレールの波うち傾向を分析することは重要です。現在の伝統的な方法は波うちの複雑な進化を捉えるのに苦労しており、この研究は、専門家の判断に大きく依存するこの傾向を正確に捉えるという課題に取り組み、自己注意（SA）、双方向の時間畳み込みネットワーク（TCN）、および双方向のゲーテッドリカレントユニット（GRU）に基づく知能予測手法を提案しています。まず、多ドメインの特徴の抽出と適応的な特徴スクリーニングを行い、最適な特徴セットを取得しました。これらの特徴は、主成分分析（PCA）とマハラノビス距離（MD）法と組み合わせて、レールの波うちの進化を反映する包括的な健康指標（CHI）を構築しました。波うちの進化中に前方と後方の情報の時間的相関を捉えるために双方向融合モデルアーキテクチャを採用し、SAをモデルに組み込んで鍵情報に焦点を当てました。その結果、双方向TCN、双方向GRU、SAを組み合わせたレール波うちの傾向予測ネットワークが作成されました。その後、多戦略改良冠ヤマアラシ最適化子（CPO）アルゴリズムが構築され、自動的に最適なネットワークのハイパーパラメータを取得しました。提案された手法は現場のレール波うちデータで検証され、他の先進的な方法と比較して優れた予測性能を示しました。要するに、提案された方法はレールの波うちの進化傾向を正確に予測することができ、現場の鉄道保守の貴重なツールとなります。",new_keywords:"マハラノビス距離、レールの波打ち、進化トレンドの予測、改良された冠予言動物最適化手法、混合型時系列ネットワーク",is_check:h},{language:"py",new_title:"Интеллектуальное прогнозирование тенденции эволюции зубчатости рельсов на основе двухстороннего TCN с самовниманием и GRU.",new_abstract:"Анализ тренда развития рельсовой волнистости с использованием обработки сигналов и глубокого обучения критичен для безопасности железнодорожного транспорта, так как текущие традиционные методы борьбы с задачей улавливания сложного развития волнистости не справляются. Настоящее исследование решает проблему точного улавливания этого тренда, который в значительной степени зависит от экспертного мнения, предлагая интеллектуальный метод прогнозирования на основе само-внимания (Self-Attention), двунаправленной временной сверточной нейронной сети (Temporal Convolutional Network, TCN) и двунаправленной управляемой рекуррентной единицы (Gated Recurrent Unit, GRU). Сначала было использовано извлечение многодоменных характеристик и адаптивное отбор признаков для получения оптимального набора признаков. Эти признаки затем были объединены с методом главных компонент и методом Махаланобиса для создания комплексного индикатора здоровья (CHI), отражающего развитие рельсовой волнистости. Была использована двунаправленная архитектура модели объединения для улавливания временных корреляций между прямой и обратной информацией во время развития волнистости, с само-вниманием, встроенным в модель для усиления фокусировки на ключевой информации. Результатом стало сеть предсказания тренда рельсовой волнистости, объединившая двунаправленную TCN, двунаправленную GRU и само-внимание. Впоследствии был создан многостратегический улучшенный оптимизатор оптимального положения (CPO) для автоматического получения оптимальных гиперпараметров сети. Предложенный метод был проверен на месте работы с данными о рельсовой волнистости, продемонстрировав превосходную предсказательную производительность по сравнению с другими передовыми методами. В итоге предложенный метод может точно прогнозировать тренд развития рельсовой волнистости, предоставляя ценный инструмент для местного технического обслуживания железной дороги.",new_keywords:"Махаланобисовское расстояние, рельсовая коррозия, прогноз тренда эволюции, улучшенный оптимизатор соснового дикобраза, гибридная сеть временных рядов.",is_check:h},{language:"sk",new_title:"자가 주의 양방향 TCN과 GRU를 기반으로한 철도 주름 발전 경향의 지능적인 예측",new_abstract:"통신프로세싱 및 딥 러닝을 사용하여 철도 주름의 진화 추세를 분석하는 것은 철도 안전을 위해 중요하며, 현재의 전통적인 방법은 주름의 복잡한 진화를 캡처하는 데 어려움을 겪고 있습니다. 본 연구는 전문가의 판단에 크게 의존하는 이 추세를 정확하게 포착하는 과제에 대응하기 위해 셀프 어텐션(SA)을 기반으로 한 지능형 예측 방법, 양방향 시간 컨볼루션 네트워크(TCN) 및 양방향 게이트 순환 유닛(GRU)을 제안함으로써 다룹니다. 먼저, 다도메인 피처 추출 및 적응형 피처 스크리닝을 사용하여 최적의 피처 세트를 얻었습니다. 이러한 피처는 주성분 분석(PCA) 및 Mahalanobis 거리(MD) 방법과 결합되어 철도 주름의 진화를 반영하는 종합 건강 지표(CHI)를 구축했습니다. 양방향 융합 모델 아키텍처를 활용하여 주름 진화 중의 순방향 및 역방향 정보 간의 시간적 상관 관계를 포착하였으며, 중요 정보에 더 집중하기 위해 모델 내에 SA가 포함되었습니다. 결과적으로, 양방향 TCN, 양방향 GRU 및 SA를 결합한 철도 주름 추세 예측 네트워크가 생겼습니다. 이후에는 최적의 네트워크 하이퍼파라미터를 자동으로 얻기 위해 다중 전략 개선된 크레스티드 쥐 메마른 최적화기(CPO) 알고리즘을 구축했습니다. 제안된 방법은 현장 철도 주름 데이터로 확인되어 다른 고급 방법들과 비교했을 때 우수한 예측 성능을 보여주었습니다. 요약하면, 제안된 방법은 철도 주름의 진화 추세를 정확하게 예측할 수 있으며 현장 철도 유지보수에 유용한 도구를 제공합니다.",new_keywords:"마할라노비스 거리, 철도 주름, 진화 추세 예측, 개선된 요철 콩고수니 옵티마이저, 혼합 시계열 네트워크",is_check:h},{language:"it",new_title:"Predizione intelligente della tendenza di evoluzione dell'ondeggiamento del binario basata su TCN bidirezionale con autoattenzione e GRU",new_abstract:"Analizzare la tendenza evolutiva della corrugazione ferroviaria utilizzando l'elaborazione del segnale e il deep learning è fondamentale per la sicurezza ferroviaria, in quanto i metodi tradizionali attuali faticano a catturare l'evoluzione complessa della corrugazione. Questo studio affronta la sfida di catturare accuratamente questa tendenza, che dipende significativamente dal giudizio esperto, proponendo un metodo di previsione intelligente basato sull'autoattenzione (SA), una rete convoluzionale temporale bidirezionale (TCN) e un'unità di ricorrenza controllata bidirezionale (GRU). In primo luogo, l'estrazione di caratteristiche multidominio e il filtraggio adattivo delle caratteristiche sono stati utilizzati per ottenere l'insieme ottimale di caratteristiche. Queste caratteristiche sono state quindi combinate con l'analisi delle componenti principali (PCA) e il metodo della distanza di Mahalanobis (MD) per costruire un indicatore di salute complessivo (CHI) che riflette l'evoluzione della corrugazione ferroviaria. È stata impiegata un'architettura di modello di fusione bidirezionale per catturare le correlazioni temporali tra informazioni in avanti e all'indietro durante l'evoluzione della corrugazione, con SA incorporata nel modello per potenziare il focus sulle informazioni chiave. Il risultato è stato una rete di previsione della tendenza della corrugazione ferroviaria che combina un TCN bidirezionale, un GRU bidirezionale e un SA. Successivamente, è stata costruita un algoritmo CPO di porcospino cresta multi-strategia migliorato per ottenere automaticamente gli iperparametri di rete ottimali. Il metodo proposto è stato convalidato con dati di corrugazione ferroviaria in loco, dimostrando prestazioni predittive superiori rispetto ad altri metodi avanzati. In sintesi, il metodo proposto può prevedere accuratamente la tendenza evolutiva della corrugazione ferroviaria, offrendo uno strumento prezioso per la manutenzione ferroviaria in loco.",new_keywords:"distanza di Mahalanobis, corrugamento del binario, previsione della tendenza evolutiva, miglioramento dell'ottimizzatore del porcospino a cresta, rete ibrida di serie temporali",is_check:h},{language:"fs",new_title:"Predicción inteligente de la tendencia de evolución de las corrugaciones en rieles basada en TCN bidireccional de autoatención y GRU.",new_abstract:"Analizar la tendencia de evolución de la corrugación de rieles utilizando procesamiento de señales y aprendizaje profundo es crucial para la seguridad ferroviaria, ya que los métodos tradicionales actuales tienen dificultades para capturar la compleja evolución de la corrugación. Este estudio aborda el desafío de capturar con precisión esta tendencia, que depende significativamente del juicio experto, proponiendo un método de predicción inteligente basado en autoatención (SA), una red convolucional temporal bidireccional (TCN) y una unidad recurrente bidireccional con compuertas (GRU). Primero, se utilizó la extracción de características multidominio y el cribado adaptativo de características para obtener el conjunto de características óptimo. Estas características se combinaron luego con el análisis de componentes principales (PCA) y el método de distancia de Mahalanobis (MD) para construir un indicador de salud integral (CHI) que refleja la evolución de la corrugación de rieles. Se empleó una arquitectura de modelo de fusión bidireccional para capturar las correlaciones temporales entre la información hacia adelante y hacia atrás durante la evolución de la corrugación, con SA incrustada en el modelo para mejorar el enfoque en la información clave. El resultado fue una red de predicción de tendencia de corrugación de rieles que combinaba un TCN bidireccional, GRU bidireccional y SA. Posteriormente, se construyó un algoritmo CPO mejorado con múltiples estrategias para obtener automáticamente los hiperparámetros de la red óptimos. El método propuesto se validó con datos de corrugación de rieles en el sitio, demostrando un rendimiento predictivo superior en comparación con otros métodos avanzados. En resumen, el método propuesto puede predecir con precisión la tendencia de evolución de la corrugación de rieles, ofreciendo una herramienta valiosa para el mantenimiento ferroviario en el sitio.",new_keywords:"distancia de Mahalanobis, corrugación ferroviaria, predicción de tendencias evolutivas, optimizador mejorado de puercoespín crestado, red de series temporales híbrida",is_check:h},{language:"po",new_title:"Previsão inteligente da tendência de evolução da corrugação ferroviária com base no TCN bidirecional de autoatenção e GRU.",new_abstract:"Analisar a tendência de evolução da corrugação ferroviária utilizando processamento de sinal e aprendizado profundo é fundamental para a segurança ferroviária, pois os métodos tradicionais atuais têm dificuldade em capturar a evolução complexa da corrugação. Este estudo aborda o desafio de capturar com precisão essa tendência, que depende significativamente do julgamento de especialistas, ao propor um método de previsão inteligente baseado em autoatenção (SA), uma rede convolucional temporal bidirecional (TCN) e uma unidade recorrente gateada bidirecional (GRU). Primeiro, a extração de características em múltiplos domínios e a triagem adaptativa de características foram usadas para obter o conjunto ideal de características. Essas características foram então combinadas com análise de componentes principais (PCA) e o método da distância de Mahalanobis (MD) para construir um indicador abrangente de saúde (CHI) que reflete a evolução da corrugação ferroviária. Uma arquitetura de modelo de fusão bidirecional foi empregada para capturar as correlações temporais entre informações para frente e para trás durante a evolução da corrugação, com a SA incorporada no modelo para aprimorar o foco em informações-chave. O resultado foi uma rede de previsão de tendência de corrugação ferroviária que combinava um TCN bidirecional, GRU bidirecional e SA. Posteriormente, um algoritmo CPO melhorado de porco-espinho com crista de múltiplas estratégias foi construído para obter automaticamente os hiperparâmetros ideais da rede. O método proposto foi validado com dados de corrugação ferroviária no local, demonstrando desempenho preditivo superior em comparação com outros métodos avançados. Em resumo, o método proposto pode prever com precisão a tendência de evolução da corrugação ferroviária, oferecendo uma ferramenta valiosa para a manutenção ferroviária no local.",new_keywords:"Distância de Mahalanobis, corrugação de trilhos, previsão de tendência de evolução, otimizador de porco-espinho com crista melhorado, rede de séries temporais híbrida.",is_check:h}]},ArtDataF:[{id:2204566,article_id:e,reference_num:h,reference:"Wang Z, Lei Z. 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Short-term wind power forecasting based on attention mechanism and deep learning. \u003Ci\u003EElectr Pow Syst Res\u003C\u002Fi\u003E 2022;206:107776.",refdoi:"https:\u002F\u002Fdx.doi.org\u002F10.1016\u002Fj.epsr.2022.107776",pubmed:a,pmc:a},{id:2204612,article_id:e,reference_num:"47",reference:"Xiang L, Liu J, Yang X, Hu A, Su H. Ultra-short term wind power prediction applying a novel model named SATCN-LSTM. \u003Ci\u003EEnerg Convers Managem\u003C\u002Fi\u003E 2022;252:115036.",refdoi:"https:\u002F\u002Fdx.doi.org\u002F10.1016\u002Fj.enconman.2021.115036",pubmed:a,pmc:a},{id:2204613,article_id:e,reference_num:bv,reference:"Yang H, Liu J, Mei G, Yang D, Deng X, Duan C. Research on real-time detection method of rail corrugation based on improved ShuffleNet V2. \u003Ci\u003EEng Appl Artif Intel\u003C\u002Fi\u003E 2023;126:106825.",refdoi:"https:\u002F\u002Fdx.doi.org\u002F10.1016\u002Fj.engappai.2023.106825",pubmed:a,pmc:a},{id:2204614,article_id:e,reference_num:"49",reference:"Li S, Mao X, Shang P, Xu X, Liu J, Qiao P. Intelligent detection of rail corrugation using ACMP-based energy entropy and LSSVM. \u003Ci\u003ENonlin Dynam\u003C\u002Fi\u003E 2023;111:8419-38.",refdoi:"https:\u002F\u002Fdx.doi.org\u002F10.1007\u002Fs11071-022-08066-2",pubmed:a,pmc:a}],ArtDataP:[{href:"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure1",post_id:"Figure1",image:"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-1.jpg"},{href:"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure2",post_id:"Figure2",image:"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-2.jpg"},{href:"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure3",post_id:"Figure3",image:"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-3.jpg"},{href:"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure4",post_id:"Figure4",image:"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-4.jpg"},{href:"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure5",post_id:"Figure5",image:"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-5.jpg"},{href:"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure6",post_id:"Figure6",image:"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-6.jpg"},{href:"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure7",post_id:"Figure7",image:"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-7.jpg"},{href:"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure8",post_id:"Figure8",image:"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-8.jpg"},{href:"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure9",post_id:"Figure9",image:"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-9.jpg"},{href:"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure10",post_id:"Figure10",image:"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-10.jpg"},{href:"\u002Farticles\u002Fir.2024.20\u002Fimage\u002FFigure11",post_id:"Figure11",image:"https:\u002F\u002Fimage.oaes.cc\u002F80d245c1-6593-4d42-8e24-7a9b32dfa757\u002Fir4020-11.jpg"}],ArtDataT:[{date_published:1731945600,section:t,section_id:w,title:"Path planning method for USVs based on improved DWA and COLREGs",doi:"10.20517\u002Fir.2024.23",abstract:"\u003Cp\u003EIn the navigation of unmanned surface vehicles (USVs), various types of obstacles may be encountered, which can be categorized into real-time collision avoidance among multiple USVs and obstacle avoidance between USVs and other obstacles. Most existing autonomous obstacle avoidance algorithms do not account for the nonlinear motion characteristics of USVs, often resulting in non-compliance with the International Regulations for Preventing Collisions at Sea (COLREGs) and a tendency to fall into local optima. To address these issues, this paper proposes a path planning algorithm that integrates the dynamic window approach (DWA) considering nonlinear characteristics with COLREGs, making the USV's motion trajectory more applicable to practical engineering scenarios. A kinematic mathematical model is established based on the motion characteristics of USVs, and an evaluation function for the optimal path is constructed using DWA. The fully informed search algorithm (FISA) is employed to select the optimal set of velocities and steering angles from the velocity sampling set, based on different cost calculation methods. The USVs use a laser radar for local obstacle detection, enabling real-time dynamic obstacle avoidance. To address the real-time collision avoidance problem among multiple USVs in open waters, the algorithm filters out COLREGs-compliant avoidance maneuvers during path planning. The correctness and feasibility of the fusion algorithm were verified through comparative simulations. In the simulated environment model, the influence of ocean currents on the USV was introduced, and multiple sets of experiments under different conditions were conducted to compare the motion trajectories, average travel distances, and average travel times of the USV. The simulation results indicate that the USV can perform accurate obstacle avoidance when encountering various types of obstacles. Compared to the traditional DWA algorithm, the proposed approach demonstrates advantages in terms of travel distance and travel time, while still achieving effective obstacle avoidance.\u003C\u002Fp\u003E",pdfurl:"https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002F2e840c3e-2dc5-4bea-bb3c-0b62e2478ad6\u002Fir4023.pdf",elocation_id:b,fpage:385,article_id:7378,viewed:d,downloaded:d,video_url:b,volume:f,year:q,tag:"385-405",image:r,authors:bP,video_img:a,journal_path:u,lpage:405,author:bP,specialissue:a,specialinfo:a,url_doi:"ir.2024.23"},{date_published:bQ,section:t,section_id:w,title:"Improved DDPG algorithm-based path planning for unmanned surface vehicles",doi:"10.20517\u002Fir.2024.22",abstract:"\u003Cp\u003EAs a promising mode of water transportation, unmanned surface vehicles (USVs) are used in various fields owing to their small size, high flexibility, favorable price, and other advantages. Traditional navigation algorithms are affected by various path planning issues. To address the limitations of the traditional deep deterministic policy gradient (DDPG) algorithm, namely slow convergence speed and sparse reward and punishment functions, we proposed an improved DDPG algorithm for USV path planning. First, the principle and workflow of the DDPG deep reinforcement learning (DRL) algorithm are described. Second, the improved method (based on the USVs kinematic model) is proposed, and a continuous state and action space is designed. The reward and punishment function are improved, and the principle of collision avoidance at sea is introduced. Dynamic region restriction is added, distant obstacles in the state space are ignored, and the nearby obstacles are observed to reduce the number of algorithm iterations and save computational resources. The introduction of a multi-intelligence approach combined with a prioritized experience replay mechanism accelerates algorithm convergence, thereby increasing the efficiency and robustness of training. Finally, through a combination of theory and simulation, the DDPG DRL is explored for USV obstacle avoidance and optimal path planning.\u003C\u002Fp\u003E",pdfurl:"https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002F69bc157a-c03e-48d7-b08c-14cc87ec711f\u002Fir4022.pdf",elocation_id:b,fpage:363,article_id:7352,viewed:d,downloaded:d,video_url:b,volume:f,year:q,tag:"363-84",image:r,authors:bR,video_img:a,journal_path:u,lpage:384,author:bR,specialissue:a,specialinfo:a,url_doi:"ir.2024.22"},{date_published:bQ,section:t,section_id:w,title:"Neurodynamics-based formation tracking control of leader-follower nonholonomic multiagent systems",doi:"10.20517\u002Fir.2024.21",abstract:"\u003Cp\u003EThis paper uses a bioinspired neurodynamic (BIN) approach to investigate the formation control problem of leader-follower nonholonomic multiagent systems. In scenarios where not all followers can receive the leader's state, a distributed adaptive estimator is presented to estimate the leader's state. The distributed formation controller, designed using the backstepping technique, utilizes the estimated leader states and neighboring formation tracking error. To address the issue of impractical velocity jumps, a BIN-based approach is integrated into the backstepping controller. Furthermore, considering the practical applications of nonholonomic multiagent systems, a backstepping controller with a saturation velocity constraint is proposed. Rigorous proofs are provided. Finally, the effectiveness of the presented formation control law is illustrated through numerical simulations.\u003C\u002Fp\u003E",pdfurl:"https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002Ff39a731b-c097-4c5a-a2b2-af4573b6f3f1\u002Fir4021.pdf",elocation_id:b,fpage:339,article_id:7351,viewed:d,downloaded:d,video_url:b,volume:f,year:q,tag:"339-62",image:r,authors:bS,video_img:a,journal_path:u,lpage:362,author:bS,specialissue:a,specialinfo:a,url_doi:"ir.2024.21"},{date_published:bT,section:"Review",section_id:935,title:"Distributed model predictive control for unmanned aerial vehicles and vehicle platoon systems: a review",doi:"10.20517\u002Fir.2024.19",abstract:"\u003Cp\u003EThis paper reviews the application of distributed model predictive control (DMPC) for autonomous intelligent systems (AIS) with unmanned aerial vehicles (UAVs) and vehicle platoon systems. DMPC is an optimal control method that formulates and solves optimization problems to adjust control strategies by predicting future states based on system models while managing constraints, and this technique has been applied to an increasing number of industrial areas. As the essential parts of AIS, UAVs and vehicle platoon systems have received extensive attention in the civil, industrial, and military fields. DMPC has the ability to quickly solve optimization problems in real-time while taking into account the prediction of the future state of the system, which fits in well with the ability of AIS to predict the environment when making decisions, so the application of DMPC in AIS has a natural advantage. This paper first introduces the basic principles of DMPC and the theoretical results in multi-agent systems (MASs). It then reviews the application of DMPC methods to UAVs and vehicle platoon systems. Finally, the challenges of the existing methods are summarized to offer insights to advance the future development of DMPC in practical applications.\u003C\u002Fp\u003E",pdfurl:"https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002Fe7dd7fcd-3c45-4298-b9e2-ec23a759ec19\u002Fir4019.pdf",elocation_id:b,fpage:293,article_id:7201,viewed:d,downloaded:d,video_url:b,volume:f,year:q,tag:"293-317",image:"https:\u002F\u002Foaepublishstorage.blob.core.windows.net\u002Fe7dd7fcd-3c45-4298-b9e2-ec23a759ec19\u002Fir4019-coverimg.jpg",authors:bU,video_img:a,journal_path:u,lpage:317,author:bU,specialissue:{id:1328,name:" Robot System Intelligentization and Application: Learning, Control and Decision"},specialinfo:a,url_doi:"ir.2024.19",image_list:"https:\u002F\u002Fimage.oaes.cc\u002Fe7dd7fcd-3c45-4298-b9e2-ec23a759ec19\u002Fir4019-coverimg.jpg"},{date_published:bT,section:t,section_id:w,title:"An in-vehicle real-time infrared object detection system based on deep learning with resource-constrained hardware",doi:"10.20517\u002Fir.2024.18",abstract:"\u003Cp\u003EAdvanced driver assistance systems primarily rely on visible images for information. However, in low-visibility weather conditions, such as heavy rain or fog, visible images struggle to capture road conditions accurately. In contrast, infrared (IR) images can overcome this limitation, providing reliable information regardless of external lighting. Addressing this problem, we propose an in-vehicle IR object detection system. We optimize the you only look once (YOLO) v4 object detection algorithm by replacing its original backbone with MobileNetV3, a lightweight feature extraction network, resulting in the MobileNetV3-YOLOv4 model. Furthermore, we replace traditional pre-processing methods with an Image Enhancement Conditional Generative Adversarial Network inversion algorithm to enhance the pre-processing of the input IR images. Finally, we deploy the model on the Jetson Nano, an edge device with constrained hardware resources. Our proposed method achieves an 82.7% mean Average Precision and a frame rate of 55.9 frames per second on the FLIR dataset, surpassing state-of-the-art methods. The experimental results confirm that our approach provides outstanding real-time detection performance while maintaining high precision.\u003C\u002Fp\u003E",pdfurl:"https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002Fd5b5c898-52d8-48b2-8229-c81553cb47df\u002Fir4018.pdf",elocation_id:b,fpage:276,article_id:7199,viewed:d,downloaded:d,video_url:b,volume:f,year:q,tag:"276-92",image:a,authors:bV,video_img:a,journal_path:u,lpage:292,author:bV,specialissue:{id:bW,name:bX},specialinfo:a,url_doi:"ir.2024.18"},{date_published:1725033600,section:t,section_id:w,title:"Parallel implementation for real-time visual SLAM systems based on heterogeneous computing",doi:"10.20517\u002Fir.2024.17",abstract:"\u003Cp\u003ESimultaneous localization and mapping has become rapidly developed and plays an indispensable role in intelligent vehicles. However, many state-of-the-art visual simultaneous localization and mapping (VSLAM) frameworks are very time-consuming both in front-end and back-end, especially for large-scale scenes. Nowadays, the increasingly popular use of graphics processors for general-purpose computing, and the progressively mature high-performance programming theory based on compute unified device architecture (CUDA) have given the possibility for large-scale VSLAM to solve the conflict between limited computing power and excessive computing tasks. The paper proposes a full-flow optimal parallelization scheme based on heterogeneous computing to speed up the time-consuming modules in VSLAM. Firstly, a parallel strategy for feature extraction and matching is designed to reduce the time consumption arising from multiple data transfers between devices. Secondly, a bundle adjustment method based solely on CUDA is developed. By fully optimizing memory scheduling and task allocation, a large increase in speed is achieved while maintaining accuracy. Besides, CUDA heterogeneous acceleration is fully utilized for tasks such as error computation and linear system construction in the VSLAM back-end to enhance the operation speed. Our proposed method is tested on numerous public datasets on both computer and embedded sides, respectively. A number of qualitative and quantitative experiments are performed to verify its superiority in terms of speed compared to other states-of-the-art.\u003C\u002Fp\u003E",pdfurl:"https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002F8983c097-e6e7-4e2a-ab70-951a0375486d\u002Fir4017.pdf",elocation_id:b,fpage:256,article_id:7145,viewed:d,downloaded:d,video_url:b,volume:f,year:q,tag:"256-75",image:r,authors:bY,video_img:a,journal_path:u,lpage:275,author:bY,specialissue:{id:bW,name:bX},specialinfo:a,url_doi:"ir.2024.17"},{date_published:1722355200,section:t,section_id:w,title:"Importance-driven denial-of-service attack strategy design against remote state estimation in multi-agent intelligent power systems",doi:"10.20517\u002Fir.2024.16",abstract:"\u003Cp\u003EThis paper introduces a novel importance-driven denial of service (IDoS) attack strategy aimed at impairing the quality of remote estimators for target agents within multi-agent intelligent power systems. The strategy features two key aspects. Firstly, the IDoS attack strategy concentrates on target agents, enabling attackers to determine the voltage sensitivity of each agent based on limited information. By utilizing these sensitivities, the proposed strategy selectively targets agents with high sensitivity to amplify disruption on the target agent. Secondly, unlike most existing denial of service attack strategies that adhere to predefined attack sequences, IDoS attacks can selectively target important packets on highly sensitive agents, causing further disruption to the target agent. Simulation results on the IEEE 39-Bus system demonstrate that, compared to existing denial of service attack strategies, the proposed IDoS attack strategy significantly diminishes the estimation quality of the target agent, confirming its effectiveness from an attacker's perspective.\u003C\u002Fp\u003E",pdfurl:"https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002F97511ed7-fdca-46a5-8b9a-e22779bd808e\u002Fir4016.pdf",elocation_id:b,fpage:244,article_id:7075,viewed:d,downloaded:d,video_url:b,volume:f,year:q,tag:"244-55",image:r,authors:bZ,video_img:a,journal_path:u,lpage:255,author:bZ,specialissue:{id:1912,name:" Performance Evaluation and Optimization for Intelligent Systems"},specialinfo:a,url_doi:"ir.2024.16"},{date_published:1722268800,section:t,section_id:w,title:"A novel fatigue driving detection method based on whale optimization and Attention-enhanced GRU",doi:"10.20517\u002Fir.2024.15",abstract:"\u003Cp\u003EFatigue driving has emerged as the predominant causative factor for road traffic safety accidents. The fatigue driving detection method, derived from laboratory simulation data, faces challenges related to imbalanced data distribution and limited recognition accuracy in practical scenarios. In this study, we introduce a novel approach utilizing a gated recurrent neural network method, employing whale optimization algorithm for fatigue driving identification. Additionally, we incorporate an attention mechanism to enhance identification accuracy. Initially, this study focuses on the driver's operational behavior under authentic vehicular conditions. Subsequently, it employs wavelet energy entropy, scale entropy, and singular entropy analysis to extract the fatigue-related features from the driver's operational behavior. Subsequently, this study adopts the cross-validation recursive feature elimination method to derive the optimal fatigue feature index about operational behavior. To effectively capture long-range dependence relationships, this study employs the gated recurrent unit neural network method. Lastly, an attention mechanism is incorporated in this study to concentrate on pivotal features within the data sequence of driving behavior. It assigns greater weight to crucial information, mitigating information loss caused by the extended temporal sequence. Experimental results obtained from real vehicle data demonstrate that the proposed method achieves an accuracy of 89.84% in third-level fatigue driving detection, with an omission rate of 10.99%. These findings affirm the feasibility of the approach presented in this study.\u003C\u002Fp\u003E",pdfurl:"https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002F5dfe9b74-3247-4d58-ae55-101930a69cd9\u002Fir4015.pdf",elocation_id:b,fpage:230,article_id:7074,viewed:d,downloaded:d,video_url:b,volume:f,year:q,tag:"230-43",image:r,authors:b_,video_img:a,journal_path:u,lpage:243,author:b_,specialissue:{id:1201,name:" Advances in Human-Assistive Technologies and Human-Robot Interactions"},specialinfo:a,url_doi:"ir.2024.15"},{date_published:1719331200,section:t,section_id:w,title:"Parameter identification of an open-frame underwater vehicle based on numerical simulation and quantum particle swarm optimization",doi:"10.20517\u002Fir.2024.14",abstract:"\u003Cp\u003EAccurate parameter identification of underwater vehicles is of great significance for their controller design and fault diagnosis. Some studies adopt numerical simulation methods to obtain the model parameters of underwater vehicles, but usually only conduct decoupled single-degree-of-freedom steady-state numerical simulations to identify resistance parameters. In this paper, the velocity response is solved by applying a force (or torque) to the underwater vehicle based on the overset grid and Dynamic Fluid-Body Interaction model of STAR-CCM+, solving for the velocity response of an underwater vehicle in all directions in response to propulsive force (or moment) inputs. Based on the data from numerical simulations, a parameter identification method using quantum particle swarm optimization is proposed to simultaneously identify inertia and resistance parameters. By comparing the forward velocity response curves obtained from pool experiments, the identified vehicle model’s mean square error of forward velocity is less than 0.20%, which is superior to the steady-state simulation method and particle swarm optimization and genetic algorithm approaches.\u003C\u002Fp\u003E",pdfurl:"https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002F16fa3eb0-55b8-4227-8f0c-aa477420de91\u002Fir4014.pdf",elocation_id:b,fpage:216,article_id:6972,viewed:d,downloaded:d,video_url:b,volume:f,year:q,tag:"216-29",image:r,authors:b$,video_img:a,journal_path:u,lpage:229,author:b$,specialissue:{id:1870,name:" Updates in Underwater Robotics"},specialinfo:a,url_doi:"ir.2024.14"},{date_published:1718899200,section:t,section_id:w,title:"Structural damage identification method based on Swin Transformer and continuous wavelet transform",doi:"10.20517\u002Fir.2024.13",abstract:"\u003Cp\u003EThe accuracy improvement of deep learning-based damage identification methods has always been pursued. To this end, this study proposes a novel damage identification method using Swin Transformer and continuous wavelet transform (CWT). Specifically, the original structural vibration data is first transferred to a time-frequency diagram by CWT, thereby capturing the characteristic information of structural damage. Secondly, the Swin Transformer is applied to learn the two-dimensional time-frequency diagram layer by layer and extract the damage information, by which the damage identification is achieved. Then, the identification accuracy of the proposed method is analyzed under various sample lengths and different levels of environmental noise to validate the robustness of this approach. Finally, the practicality of this method is verified through laboratory test. The results show the proposed method can effectively recognize the damage and achieve excellent accuracy even under noise interference. Its accuracy reaches 99.6% and 99.0% under single damage and multiple damage scenarios, 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permanently:",url:"https:\u002F\u002Fwww.portico.org\u002Fpublishers\u002Foae\u002F",img:"https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20230911\u002F67d78ebf8c55485db6ae5b5b4bcda421.jpg"},follow:[{title:"LinkedIn",url:"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fintelligence-robotics\u002F",icon:"icon-linkedin"},{title:"Twitter",url:ka,icon:"icon-tuite1"}],wechat_img:a,twitter:{url:ka,img:"https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20230824\u002F5249ddabb6d642558c9843fba9283219.png"}},top:{path:N,pid:D,journal_img:bz,journal_name:C,mpt:"40 days",issn:j$,indexing:{ESCI:"https:\u002F\u002Fwww.oaepublish.com\u002Fnews\u002Fir.852",Scopus:aP,"Google Scholar":"https:\u002F\u002Fscholar.google.com.hk\u002Fcitations?view_op=list_works&hl=zh-CN&hl=zh-CN&user=-Hx5OVYAAAAJ",Dimensions:"https:\u002F\u002Fapp.dimensions.ai\u002Fdiscover\u002Fpublication?and_facet_source_title=jour.1423782",Lens:"https:\u002F\u002Fwww.lens.org\u002Flens\u002Fsearch\u002Fscholar\u002Flist?p=0&n=10&s=date_published&d=%2B&f=false&e=false&l=en&authorField=author&dateFilterField=publishedYear&orderBy=%2Bdate_published&presentation=false&preview=true&stemmed=true&useAuthorId=false&publicationType.must=journal%20article&sourceTitle.must=Intelligence%20%26%20Robotics&publisher.must=OAE%20Publishing%20Inc."},editor:bA,journal_rank:a,journal_flyer:"https:\u002F\u002Ff.oaes.cc\u002Findex_ad\u002Fflyer\u002FIR-flyer.pdf",qksearch:["Intelligence","Robotics","Reinforcement Learning","Machine Learning","Unmanned Vehicles","UAV"],sitetag:"Intell Robot",ad:[],colour_tag:by,score:a,mobile_top_img:a,impact_factor:[{factor:bD,url:aP},{factor:i,url:a}],rgba:bB,log_image:bC},webinfo:{},searchKey:a,loading:O,appid:a,videoPlay:{show:bE,href:a}},editer:{editList:{list:{}}},userdata:{showLogin:bE,logined:bE}},serverRendered:O,routePath:"\u002Farticles\u002Fir.2024.20",config:{_app:{basePath:kb,assetsPath:kb,cdnURL:"https:\u002F\u002Fg.oaes.cc\u002Foae\u002Fnuxt\u002F"}}}}("",null,1,0,7277,4,"2021","1","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240506\u002Fea3d9071c35b4bf3982ffe25f1083620.png","2",3,"3",2,5,"4","5",2024,"0","6","Research Article","IR","[]",927,"https:\u002F\u002Fmjl.clarivate.com\u002Fsearch-results","2022","2017","33",10,"Intelligence & Robotics",40,15,20,"Lens",34,6,7,35,"2020",32,"ir",true,"39",9,"Extracellular Vesicles and Circulating Nucleic Acids","Microbiome Research Reports","One Health & Implementation Research","Chemical Synthesis","Energy Materials","Journal of Materials Informatics","Microstructures","Minerals and Mineral Materials","Soft Science","Complex Engineering Systems","Disaster Prevention and Resilience","Green Manufacturing Open","Journal of Smart Environments and Green Computing","Journal of Surveillance, Security and Safety","Ageing and Neurodegenerative Diseases","Artificial Intelligence Surgery","Cancer Drug Resistance","Connected Health And Telemedicine","Hepatoma Research","Journal of Cancer Metastasis and Treatment","Journal of Translational Genetics and Genomics","Metabolism and Target Organ Damage","Mini-invasive Surgery","Plastic and Aesthetic Research","Rare Disease and Orphan Drugs Journal","The Journal of Cardiovascular Aging","Vessel Plus","Carbon Footprints","Journal of Environmental Exposure Assessment","Water Emerging Contaminants & Nanoplastics","40","7","8","9","11",23,65,22,14,25,30,46,8,24,"ESCI, CAS, Dimensions, Lens, CNKI",17,11,55,12,"2015","ESCI, CAS, Scopus, Wanfang Data, CNKI, Dimensions, Embase",13,"https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101199351",90,39,16,"Journal of Unexplored Medical Data",18,21,26,"ESCI, Scopus, CAS, Dimensions, Lens, CNKI","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240809\u002Fccc8ac23b789404781787655e4495af4.png",117,33,"Neuroimmunology and Neuroinflammation","2014",29,31,"ISSN XXXX-XXXX (Coming soon)","Soil Health","Space Mission Planning & Operations","Stomatological Disease and Science","ISSN 2770-3541 (Online)","10","14","15","16","20","25","28","29","30","36","37","42","44","48",70,"Contact Us","#0047bb","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231121\u002F59802903b17e4eebae240e004311d193.jpg","Simon X. Yang","rgb(0,71,187)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fir.png","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240813\u002F49390c7e86ab40a58ee862e8c1af65ba.png",false,"Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU","\u003Cp\u003EAnalyzing the evolution trend of rail corrugation using signal processing and deep learning is critical for railway safety, as current traditional methods struggle to capture the complex evolution of corrugation. This present study addresses the challenge of accurately capturing this trend, which relies significantly on expert judgment, by proposing an intelligent prediction method based on self-attention (SA), a bidirectional temporal convolutional network (TCN), and a bidirectional gated recurrent unit (GRU). First, multidomain feature extraction and adaptive feature screening were used to obtain the optimal feature set. These features were then combined with principal component analysis (PCA) and the Mahalanobis distance (MD) method to construct a comprehensive health indicator (CHI) that reflects the evolution of rail corrugation. A bidirectional fusion model architecture was employed to capture the temporal correlations between forward and backward information during corrugation evolution, with SA embedded in the model to enhance the focus on key information. The outcome was a rail corrugation trend prediction network that combined a bidirectional TCN, bidirectional GRU, and SA. Subsequently, a multi-strategy improved crested porcupine optimizer (CPO) algorithm was constructed to automatically obtain the optimal network hyperparameters. The proposed method was validated with on-site rail corrugation data, demonstrating superior predictive performance compared to other advanced methods. In summary, the proposed method can accurately predict the evolution trend of rail corrugation, offering a valuable tool for on-site railway maintenance.\u003C\u002Fp\u003E","https:\u002F\u002Fv.oaes.cc\u002Fuploads\u002F20241021\u002F6879d4f4c749472a807548446dea5a90.mp4","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20241021\u002Fd5f21a891ccd43739a59e3f817c67e42.png","Liu J H, Yang W H, He J, Wang Z M, Jia L, Zhang C F, Yang W W. Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU. \u003Ci\u003EIntell Robot\u003C\u002Fi\u003E 2024;4(4):318-38. http:\u002F\u002Fdx.doi.org\u002F10.20517\u002Fir.2024.20","Liu J H, Yang W H, He J, Wang Z M, Jia L, Zhang C F, Yang WW. Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU. \u003Ci\u003EIntell Robot\u003C\u002Fi\u003E. 2024;4:318-38. http:\u002F\u002Fdx.doi.org\u002F10.20517\u002Fir.2024.20","Liu, Jian-Hua, Chang-Fan Zhang, and Wei-Wei Yang. 2024. \"Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU\" \u003Ci\u003EIntell Robot\u003C\u002Fi\u003E. 4, no.4: 318-38. http:\u002F\u002Fdx.doi.org\u002F10.20517\u002Fir.2024.20","Liu, J. H.; Yang, W. H.; He, J.; Wang, Z. M.; Jia, L.; Zhang, C. F.; Yang, W. W. Intelligent prediction of rail corrugation evolution trend based on self-attention bidirectional TCN and GRU. \u003Ci\u003EIntell. Robot.\u003C\u002Fi\u003E \u003Cb\u003E2024\u003C\u002Fb\u003E, \u003Ci\u003E4\u003C\u002Fi\u003E, 318-38. http:\u002F\u002Fdx.doi.org\u002F10.20517\u002Fir.2024.20","12","13","Shiqi Liu, ... Eugene Levin",1731427200,"Menglong Hua, ... Zihao Chen","Xiao-Wen Zhao\u003Ca href='https:\u002F\u002Forcid.org\u002F0000-0001-7873-8708' target='_blank'\u003E\u003Cimg src='https:\u002F\u002Fi.oaes.cc\u002Fimages\u002Forcid.png' class='author_id' alt='Xiao-Wen Zhao'\u003E\u003C\u002Fa\u003E, ... Zhi-Wei Liu",1727107200,"Yang Peng\u003Ca href='https:\u002F\u002Forcid.org\u002F0009-0008-0113-850X' target='_blank'\u003E\u003Cimg src='https:\u002F\u002Fi.oaes.cc\u002Fimages\u002Forcid.png' class='author_id' alt='Yang Peng'\u003E\u003C\u002Fa\u003E, ... Yunkai Lv\u003Ca href='https:\u002F\u002Forcid.org\u002F0000-0001-5212-8629' target='_blank'\u003E\u003Cimg src='https:\u002F\u002Fi.oaes.cc\u002Fimages\u002Forcid.png' class='author_id' alt='Yunkai Lv'\u003E\u003C\u002Fa\u003E","Tingting Zhuang, ... Xiaoyu Tang\u003Ca href='https:\u002F\u002Forcid.org\u002F0000-0002-6038-9623' target='_blank'\u003E\u003Cimg src='https:\u002F\u002Fi.oaes.cc\u002Fimages\u002Forcid.png' class='author_id' alt='Xiaoyu Tang'\u003E\u003C\u002Fa\u003E",1994," Recent Advances in Embodied Artificial Intelligence","Han Liu, ... Tingting Lv","Xia Zhao, ... Lei Li","Zuojin Li\u003Ca href='https:\u002F\u002Forcid.org\u002F0000-0002-8154-3968' target='_blank'\u003E\u003Cimg src='https:\u002F\u002Fi.oaes.cc\u002Fimages\u002Forcid.png' class='author_id' alt='Zuojin Li'\u003E\u003C\u002Fa\u003E, ... Dongyang Li","Mingzhi Chen, ... Kaimin Ji","Jingzhou Xin\u003Ca href='https:\u002F\u002Forcid.org\u002F0000-0001-7494-298X' target='_blank'\u003E\u003Cimg src='https:\u002F\u002Fi.oaes.cc\u002Fimages\u002Forcid.png' class='author_id' alt='Jingzhou Xin'\u003E\u003C\u002Fa\u003E, ... Chenglong Xiang","About","Editorial Policies","Journals","Academic Talks","Biology & Life Science","Chemistry & Materials Science","Computer Science & Engineering","\u002Fir","Medicine & Public Health","Environmental Science","#837fbc","ISSN 2769-5301 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231218\u002F6135b005a8674b878a7d1a5c91ac9869.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fand","Wei-Dong Le","CAS, Dimensions, Lens, CNKI","rgb(99,94,171)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fand.jpg","and","#030072","ISSN 2771-0408 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002F14bc775310574f91b457ff09d6c6d950.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fais","Andrew A. Gumbs","ESCI, Scopus, Dimensions, Lens","rgb(3,0,114)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fais.png","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240813\u002Fb7358d0632aa413fa84630ef0a2fb7a0.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101189039","ais",83,"#e8b475","ISSN 2578-532X (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231113\u002F9db4b73f2e954900807499f00094fcf0.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fcdr","Godefridus J. (Frits) Peters","2018","337","ESCI, PMC, Scopus, CAS, CNKI, Dimensions, Lens, Embase","rgb(224,155,71)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fcdr.jpg","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240104\u002F502e35965fc448e0bc9f2d6c1b0bb3d7.png","https:\u002F\u002Fwww.ncbi.nlm.nih.gov\u002Fpmc\u002Fjournals\u002F4180\u002F","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240611\u002F5def3ab5829743ce8d55e664cabe0dcb.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101047803","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240620\u002Fed79fcbfce8e4cd38a36cc4d5f7d473d.png","https:\u002F\u002Fjcr.clarivate.com\u002Fjcr-jp\u002Fjournal-profile?journal=CANCER%20DRUG%20RESIST&year=2022","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240620\u002F86311aafb7a945a98a7ee95c3627173b.png","cdr",418,38,"#1d57a5","ISSN 2770-6249 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fimages_2018\u002Foae\u002Fcover_comengsys.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fcomengsys","Hamid Reza Karimi","Scopus, Dimensions, Lens","rgb(29,87,165)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fces.png","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240722\u002F4748814aab87441d809e5b9ddcd9d56e.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101189038","comengsys",71,58,"#75ac9d","ISSN 2993-2920 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231218\u002F0b7a80c0f6e545818e143ea48fe9be90.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fchatmed","Yuan-Ting Zhang","rgb(117, 172, 157)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fchatmed.jpg","chatmed","#00a5b3","ISSN 2769-5247 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231121\u002F1598f33bf2284642830511e489eb31bf.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fcs","Bao-Lian Su","67","Scopus, CAS, Dimensions, Lens, CNKI","rgb(0,165,179)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fcs.jpg","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240813\u002F6d91bea2ff474cb9b7ec10017b26ec2a.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101189036","cs",157,51,"#6cc24a","ISSN 2831-932X (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002Fb1a3c337f6a54bcb945e0d9ab6afe0ec.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fcf","Yong Geng","rgb(86,164,55)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fcf.jpg","cf","#01588b","ISSN 2832-4056 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002Fe5808de306274ba287f6f24bf45eb910.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fdpr","Jie Li","rgb(1,88,139)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fdpr.jpg","dpr",41,"#008c15","ISSN 2770-5900 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002Fe1df1779a10c4a7486e89c5f5c4994f3.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fenergymater","Yuping Wu","92","rgb(0,140,21)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fem.jpg","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240621\u002Fab41932db942489c841586b032349a31.png","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240625\u002F3f1358bb24954c8eae11907a77ab8b18.png","energymater",196,"#b15ee9","ISSN 2767-6641 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002Fb7fc9d6a51bf4adda4ff6dec5002690b.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fevcna","Yoke Peng Loh","68","ESCI, Scopus, CAS, CNKI, Dimensions, Lens","rgb(177,94,233)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fevcna.jpg","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240228\u002Ff15fa97315a24fcc8bc2910eed05ec8a.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101160588","https:\u002F\u002Fmjl.clarivate.com:\u002Fsearch-results?issn=2767-6641&hide_exact_match_fl=true&utm_source=mjl&utm_medium=share-by-link&utm_campaign=search-results-share-this-journal","evcna",131,56,"#1d6960","ISSN 2835-7590 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002Fb95b79391630482d83323d705e486a35.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fgmo","Hongchao Zhang","rgb(29,105,96)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fgmo.png","gmo","#f3906d","ISSN 2454-2520 (Online)\u003Cbr\u003EISSN 2394-5079 (Print)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002F4ef51b2e673a40ca9ebf0a414f21adb5.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fhr","Guang-Wen Cao","465","rgb(239,108,62)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fhr.jpg","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240313\u002Fee358186a06449d5b2c8ea15f712cbed.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101058282","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240625\u002Faf5ee9028a554c6aa6f0030bbcbfe55d.png","https:\u002F\u002Fjcr.clarivate.com\u002Fjcr-jp\u002Fjournal-profile?journal=HEPATOMA%20RES&year=2023","hr",543,"https:\u002F\u002Fwww.oaepublish.com\u002Fir","ESCI, Scopus, Google Scholar, Dimensions, Lens",60,"#44b762","ISSN 2454-2857 (Online)\u003Cbr\u003EISSN 2394-4722 (Print)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231121\u002Fe46043c3fcaf49f38eef7654adff79e1.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fjcmt","476","rgb(68,183,98)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fjcmt.jpg","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231201\u002F8b13deaaf6e24f9189ca91e62e3b84ab.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101058912","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240621\u002F38a515f5996349019b5018be997f4446.png","https:\u002F\u002Fmjl.clarivate.com\u002F","jcmt",527,"#00629b","ISSN 2770-372X (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002F4fb1d2ad901a4d94945ad58e1a208c41.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fjmi","Tong-Yi Zhang","rgb(0,98,155)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fjmi.jpg","jmi",81,"#5aa8d9","ISSN 2578-5281 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002Fe326d72dd71546d193c9f1e77d5aea9c.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fjtgg","Sanjay Gupta, Andrea L. 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Lens","rgb(123,164,219)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fmrr.jpg",101,"https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240104\u002F206342e31750460bb7449112e6aafba6.png","https:\u002F\u002Fwww.ncbi.nlm.nih.gov\u002Fpmc\u002Fjournals\u002F4522\u002F","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240621\u002F1dfbaaeea68e40c98d7494b8c702b695.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101195678?origin=resultslist","https:\u002F\u002Fwww.oaepublish.com\u002Fnews\u002Fmrr.836","mrr","#6f7bd4","ISSN 2770-2995 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231113\u002F1738c68b2d0a41edae268adf4ea5c7e2.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fmicrostructures","Shujun Zhang","53","ESCI, Scopus, CAS, Dimensions, 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Dimensions","rgb(41,167,168)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fmis.jpg","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240208\u002F6020c4a092dd44578a776056b5057abc.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101115668","mis",395,27,"#33a7d9","ISSN 2349-6142 (Online)\u003Cbr\u003EISSN 2347-8659 (Print)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002F026484b9f2b042fda9b7afffc0d9f62e.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fneurosciences","296","rgb(51,167,217)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fnn.jpg",104,"neurosciences",296,28,"#43b02a","ISSN 2769-6413 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002Ff65f06a7eae24d1484bad38ff0a2bef2.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fohir","Dimensions","Jose M. Martin-Moreno","rgb(67,176,42)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fohir.jpg",105,"ohir","#c45284","ISSN 2349-6150 (Online)\u003Cbr\u003EISSN 2347-9264 (Print)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002F8d72c3f669d5490ea3b9abc424d66cb6.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fpar","Wen-Guo Cui","520","rgb(196,82,132)","ESCI, Scopus, Dimensions, CNKI, Lens, Wanfang","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fpar.jpg","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240607\u002F9bbbcad710cf457e85d63b1811faefb1.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101111784","https:\u002F\u002Fclarivate.com\u002Fproducts\u002Fscientific-and-academic-research\u002Fresearch-discovery-and-workflow-solutions\u002Fwebofscience-platform\u002Fweb-of-science-core-collection\u002Femerging-sources-citation-index\u002F","par",620,"#d62598","ISSN 2771-2893 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231121\u002F6ee75e8ab583446b90c97865c2c68464.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Frdodj","Daniel Scherman","Dimensions, Lens","rgb(214,37,152)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Frdodj.jpg",106,"rdodj",78,"#1f4e79","ISSN 2769-5441 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002F98e2ed5023ee466c8890d749bfd97597.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fss","YongAn Huang","58","ESCI, Scopus, CAS, Lens, Dimensions, CNKI","rgb(31,78,121)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fss.jpg",107,"https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240606\u002Fcfa200290630438c8e3a3cae1536af37.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101178114","ss",54,"#a96318","https:\u002F\u002Fi.oaes.cc\u002Fimages_2018\u002Foae\u002Fcover_sh.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fsh","Manoj Shukla","rgb(169,99,24)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fsh.jpg",108,"sh","#4475e1","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231218\u002F79ccda0bc909441587a0b70e0a8ef1cc.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fsmpo","Madjid Tavana","rgb(68,117,225)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fsmpo.jpg",109,"smpo","#53be9b","ISSN 2573-0002 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231218\u002F54ba11ec5c57448a84169c318bb94fd4.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fsds","Nikolaos G. Nikitakis","rgb(62,163,130)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fsds.jpg",110,"sds","#0038a0","ISSN 2768-5993 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231113\u002F695c0f0de82c43f189957ff572f0798c.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fjca","89","rgb(0,56,160)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fjca.png","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240611\u002Fd7d617bceee246428da2ed07a84efd5e.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101176723","https:\u002F\u002Fwww.oaepublish.com\u002Fnews\u002Fjca.838","jca",129,36,"#db6868","ISSN 2574-1209 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002Fe64051237c8a454085bb02abfdc6dd12.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fvp","290","CAS, Scopus, CNKI, Dimensions, Lens","rgb(219,104,104)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fvp.jpg","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240606\u002Fa5cb40102c2d4ec4b1a362fe1f27d324.jpg","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101064905","vp",351,"5555","\u002Fir\u002Fcontact_us","Video Abstract Guidelines","\u002Fir\u002Fvideo_abstract_guidelines","2770-3541 (Online)","https:\u002F\u002Ftwitter.com\u002FOAE_IR","\u002F"));</script><script src="https://g.oaes.cc/oae/nuxt/b06ddfb.js" defer></script><script src="https://g.oaes.cc/oae/nuxt/d1923ac.js" defer></script><script src="https://g.oaes.cc/oae/nuxt/0a3b980.js" defer></script><script src="https://g.oaes.cc/oae/nuxt/3e8004d.js" defer></script><script src="https://g.oaes.cc/oae/nuxt/b19d7ea.js" defer></script> </body> </html> <div id="noIe" style="display: none;"> <style> #noIe { background: rgba(99, 125, 255, 1); width: 100%; height: 100vh; position: fixed; top: 0; left: 0; z-index: 999999; } #noIe .container 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padding-top: 100px; } #noIe .logo { height: 24px; } @media screen and (max-width: 820px) { #noIe .container { width: 432px; left: 50%; margin-left: -216px; margin-top: 48px; position: relative; top: 0; } #noIe .container ul { width: 290px; height: 352px; margin-left: -30px; margin-top: 40px; } #noIe .container li { margin-left: 30px; margin-top: 24px; } #noIe .logo-container { padding-top: 121px; padding-bottom: 20px; } } </style> <div class="container"> <p class="title">The current browser is not compatible</p> <p class="title2">The following browsers are recommended for the best use experience</p> <ul> <li> <a href="https://www.google.cn/chrome/" target="_blank"> <img src="https://gw.alicdn.com/imgextra/i2/O1CN01Nn0IoE1cmXZ6gFiM3_!!6000000003643-2-tps-230-230.png" /> <p>Chrome</p> </a> </li> <li> <a href="http://www.firefox.com.cn/" target="_blank"> <img src="https://gw.alicdn.com/imgextra/i3/O1CN01P8aqdX1HdHczGialK_!!6000000000780-2-tps-230-230.png" /> <p>Firefox</p> </a> </li> <li> <a href="https://www.apple.com/safari/" target="_blank"> <img src="https://gw.alicdn.com/imgextra/i4/O1CN01vVxDF11chVD0nsbiZ_!!6000000003632-2-tps-230-230.png" /> <p>Safari</p> <p class="tip">Only supports Mac</p> </a> </li> <li> <a href="https://www.microsoft.com/zh-cn/edge" target="_blank"> <img src="https://gw.alicdn.com/imgextra/i4/O1CN01UW7hs31Xa6jfm2a2O_!!6000000002939-2-tps-230-230.png" /> <p>Edge</p> </a> </li> </ul> <div class="logo-container"> <img 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6+pEPbLVtqaLkoNUjKoLW15AqDlqlWvQLAHoCcPAiLY1oKiO9yUNTgSAC3n6ZyfLsolmAYAhs1F4sSxnm3VYAyp8cFFrB7Dh4qDzPJiNvLqeCkOJ0y7aAP41LmbylyUJSATAOGGpnbLA7gFN/XUMYBWc5FMSxjGtsVB116CILUcKxd1HnQQ+tCSN+igB+ADybkt1HrDvyeg9N2speYAPmwukjBHQFUtSwWar7FShZThIEqMB+gqoIoBs/bAh9/bgJx12Dw0jC8B5cZFJQDqzUUyzPmrWo2nGZSL5dqQsrB4c8APB/UaUH1sdPjGHX/WjG+LgHLiok4BhHmnd/JQwKpsoF4sV4PU/JNK3WgGavj88k1J/cYOcnorSLVZJH8xeBuGLuq0uUhargrjOYF4fHARmzrEdZ2UhX08LcgrzS2++84re/osHoePMekPiCf7ckVzkXhRrhAQ7xUsfmqQ0p7p53C/qnpVNzUFcGKW8mVP9Qp5UxXIRQVGE8EM8OjVZnb1opODi9g8jV0ZUkapXs/nNM/Rxb8Cv8VNfopJfwrKGti+pHg426pFRumsabFcy0lpD6K5qy+t6KZSZIVRTdVkWYcFpCJQlpkAAHzWXCQjj5yiVWbSdgVinyA1KJiLenHeBscMAfUXN9BFLenR/74D9Os0c88mgh5stqqdu1xO4ROkekW82aXIP9M85hf/eOe3BjdXIeNrpV9++43FYSOP0rw6bNbYOa8brwQpqUdpDqLEYtOwQ6na+68vHv1KvX7AJQevQEp948C8uUh8mgiszqXnes3Xqk5KOxUZFvyeVz3/5J/vaHdeGIC6pvf/87woUFjFRYWwWXIway4S5+nsqpCqElLX5LtLKXp81fUL/XTPp4nA6lxaeXwZL5yUqxYsVpJUVS1/+MPXP9M9QaZ612dYfReV+NLV4Gyr1obNkoPzvJ5argqpQPEcxiW5930FQVniqyZ5slc6FyVLDizSsVwXWftSkyqDUoagGDJ4sjfy5KtFsFly0MvTKa4KKfaANnRSf/ybarpHeNpq3lwkubvos61aFTavksulWK4BqTLOQmVVwBBc10+/+7aMab5VytnK+4s94C1LbRyk/vvvUk2yhksOTn3Y4kNIUVTx1bNIY+FJRxJCiqIK7KSk9bHFa9e82eJDSFFUQSVLDixcVOLTFh9CiqKKqzZsnrS3ffqSPkCqynuNot7YRVVh86LPU1/6YRFS1Mbq+YOHZbh/LdIxL9t3M93zUD9/9C8GwVBf/uBRoSElSw4serN7UyzXgNSct7qd3vsRIVUghTl8ZmRwzMSzflhrQypWPIc93uemChiC65rqOimn968sObD4zJav12tVSKlawu+9XnzDZ9TfvPuV5nnVQF3T8wdvaYOj4QhQgZGL+syH/Yc+OymgHF0V1L6Derpn9w6/Quovb6s/tW84OvU29HtF+fbyCG8hVehBJG8dVrvz33v0TVEHUWH097fe1jzcobgcSxdVhc2Tt7aPxXIfIVX0QaSaz//63a/GjK+tpj985PU9cIMi6C/cHDcXycD3a7USpLZ/dzmF7hO+HXEjRZU2BGIAiWZ8sXtEUNmmfG3jU64qHy8pyuS1zqq2EXTXarTh8ROGW1K90KBOMJJfa8rxHRJPmf789o+1D7lztlVrGTqTAfQ2M08BDH1P865Uuby8XOk/zj+ptAF8qnw+HxTtpQxpFyPoPxL+SfD7oxDA58rHfYxJf1SY4GYF/2dWhz/7659myhPMDEC9KINfQ6vywzzdE1nMyr0CuihtQCVBBykmfYv4RvRQNzpWvbTaw20lRdfKkJK6VKJ8Pntpt1BFXotBv5wunGvHl7Up84n2WF5xTnngpF4dUGqDNO36v0paYLpnPHAs4tvD7lHAWx9oLpIhbLZ4DRhdfyBlMRNt+36RBaIW5ziWF41mylI+7UG0U7S02lgW17F2tlVjjH2AlKR85wbntZ92vc7tR7BpNjZwNIgOsXvU4u1v6nqeyD47KmcnBcNZ+dO069+ShLSLAWz2w82Czo0Dxiq+T7F7VGfKl8Swe8vzU4LKA0gFHYygX0B/cZF9ApUA6tDpjD7pW7nVzBESVJYTAUHliZMyv8h5gyrtIki7GBoCan5HDK3iu01QvSigzwiqEkNK0hTTi5x281njI2uhYth0QrxSFHRuaX+TLcC0SkmuQBVu+Fiwvr+enm3VBtYbkQmp22U9UxynXUxduaq0i1BWkz+D/raXZc2Czr2ckuX33gbwDLtH0aYuT5DtLGPjjzkEEEv7X+qeqqy7rL1SqSwP7BHcdCqcIVv+MAo6essgpPleQ4DgqlncY6nr3a3do4Fhyrkc2x6AodTD8pPxthhM+pXlPwo8njn6dmNkdchhkbfRuNgWow2pKoAvc4hVgmzTZIysa2h8z/9XR9ZeN0S2y3zH8XmfBp03cEiZy5nCZvnD6+KaymdeAWv593dBYLTWpzuGlICqB+BJDvdvYUH1iqYAooOLWG2CU4WUgCqCzfvAyqYZgPqttaibB24D+huPnUGgAJByPRGUUXMArYOLWCXLUX+lVdBBBLslCWVS640BlQ2sIeyWJGy8JPVqMBJraRuK7bSt3rvXAF97dZtO7l2Heg3gGF9TUI0AfMZI+CETSMn+szbDe6POxW2uk6ZwtrcHVZsZQYkhJaAaADhhiK9pBq2lBFlR+iOG1FQhbNf/UXlCSkAVAThlmAFJzxor1aFeDypOBLZu6sqxMrUuK6QEVC2CCkBWKI/VjzrpcyKwBVUsjoqgKiukCCoAwEeai05vABUnAoKKkCKo1gLUwPxTCCqCipBSA9WmPNqdOwPUdVCxmG4Lqir41K+8kBJQtTdgIM0BhE4B9RJUAwAfcsY3A1UqjoqutayQElANAHyAcj7eTQBUc31/YLYqPeSMbweq5iJpcTIoMaQEVDGypfNl2uJxEnRW2I9nA6oYk34dXDltCauhpH90VWWElIAqDTpoyIxUZFeVIHv7cuTdmU36bQCPwUWJ1q7qMez7URFSOcJqKK6qaAsT5wA+FvcUe3uWk/4Ik35V4sv0xAZWo+YiCWXCJazKBqklVxUBeF9SFJ8H00wGfPWeXTV9gVUk6QlhZZgCCqweMw3UkXo/KS3JCzgb8rPvQazmyLqBDk0XZrpU1puqhazIrt8/qYD9pLQl/amu7mObOPunMYD2wUWskl14C6kboBXKRa6LG6g5CPQUWZfPkdfpnA6wluMbQKMNNCF1E7TqEuP6UqxrJbmLxsi6co68clIURVGWesAQUBRFSFEURRFSFEURUhRFUYQURVEUIUVRFCFFURRFSFEURUhRFEURUhRFUYQURVHF0/8HAO0wc5gipK7rAAAAAElFTkSuQmCC" 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