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<!DOCTYPE html> <html lang="en" class="no-js"> <head> <meta charset="UTF-8"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <meta name="applicable-device" content="pc,mobile"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="robots" content="max-image-preview:large"> <meta name="access" content="Yes"> <meta name="robots" content="noindex"> <meta name="360-site-verification" content="1268d79b5e96aecf3ff2a7dac04ad990" /> <title>Existence of reservoir with finite-dimensional output for universal reservoir computing | Scientific Reports</title> <meta name="journal_id" content="41598"/> <meta name="dc.title" content="Existence of reservoir with finite-dimensional output for universal reservoir computing"/> <meta name="dc.source" content="Scientific Reports 2024 14:1"/> <meta name="dc.format" content="text/html"/> <meta name="dc.publisher" content="Nature Publishing Group"/> <meta name="dc.date" content="2024-04-11"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="dc.language" content="En"/> <meta name="dc.copyright" content="2024 The Author(s)"/> <meta name="dc.rights" content="2024 The Author(s)"/> <meta name="dc.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="dc.description" content="In this paper, we prove the existence of a reservoir that has a finite-dimensional output and makes the reservoir computing model universal. Reservoir computing is a method for dynamical system approximation that trains the static part of a model but fixes the dynamical part called the reservoir. Hence, reservoir computing has the advantage of training models with a low computational cost. Moreover, fixed reservoirs can be implemented as physical systems. Such reservoirs have attracted attention in terms of computation speed and energy consumption. The universality of a reservoir computing model is its ability to approximate an arbitrary system with arbitrary accuracy. Two sufficient reservoir conditions to make the model universal have been proposed. The first is the combination of fading memory and the separation property. The second is the neighborhood separation property, which we proposed recently. To date, it has been unknown whether a reservoir with a finite-dimensional output can satisfy these conditions. In this study, we prove that no reservoir with a finite-dimensional output satisfies the former condition. By contrast, we propose a single output reservoir that satisfies the latter condition. This implies that, for any dimension, a reservoir making the model universal exists with the output of that specified dimension. These results clarify the practical importance of our proposed conditions."/> <meta name="prism.issn" content="2045-2322"/> <meta name="prism.publicationName" content="Scientific Reports"/> <meta name="prism.publicationDate" content="2024-04-11"/> <meta name="prism.volume" content="14"/> <meta name="prism.number" content="1"/> <meta name="prism.section" content="OriginalPaper"/> <meta name="prism.startingPage" content="1"/> <meta name="prism.endingPage" content="10"/> <meta name="prism.copyright" content="2024 The Author(s)"/> <meta name="prism.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="prism.url" content="https://link.springer.com/articles/s41598-024-56742-7"/> <meta name="prism.doi" content="doi:10.1038/s41598-024-56742-7"/> <meta name="citation_pdf_url" content="https://www.nature.com/articles/s41598-024-56742-7.pdf"/> <meta name="citation_fulltext_html_url" content="https://link.springer.com/articles/s41598-024-56742-7"/> <meta name="citation_journal_title" content="Scientific Reports"/> <meta name="citation_journal_abbrev" content="Sci Rep"/> <meta name="citation_publisher" content="Nature Publishing Group"/> <meta name="citation_issn" content="2045-2322"/> <meta name="citation_title" content="Existence of reservoir with finite-dimensional output for universal reservoir computing"/> <meta name="citation_volume" content="14"/> <meta name="citation_issue" content="1"/> <meta name="citation_online_date" content="2024/04/11"/> <meta name="citation_firstpage" content="1"/> <meta name="citation_lastpage" content="10"/> <meta name="citation_article_type" content="Article"/> <meta name="citation_fulltext_world_readable" content=""/> <meta name="citation_language" content="en"/> <meta name="dc.identifier" content="doi:10.1038/s41598-024-56742-7"/> <meta name="DOI" content="10.1038/s41598-024-56742-7"/> <meta name="size" content="669321"/> <meta name="citation_doi" content="10.1038/s41598-024-56742-7"/> <meta name="citation_springer_api_url" content="http://api.springer.com/xmldata/jats?q=doi:10.1038/s41598-024-56742-7&api_key="/> <meta name="description" content="In this paper, we prove the existence of a reservoir that has a finite-dimensional output and makes the reservoir computing model universal. Reservoir computing is a method for dynamical system approximation that trains the static part of a model but fixes the dynamical part called the reservoir. Hence, reservoir computing has the advantage of training models with a low computational cost. Moreover, fixed reservoirs can be implemented as physical systems. Such reservoirs have attracted attention in terms of computation speed and energy consumption. The universality of a reservoir computing model is its ability to approximate an arbitrary system with arbitrary accuracy. Two sufficient reservoir conditions to make the model universal have been proposed. The first is the combination of fading memory and the separation property. The second is the neighborhood separation property, which we proposed recently. To date, it has been unknown whether a reservoir with a finite-dimensional output can satisfy these conditions. In this study, we prove that no reservoir with a finite-dimensional output satisfies the former condition. By contrast, we propose a single output reservoir that satisfies the latter condition. This implies that, for any dimension, a reservoir making the model universal exists with the output of that specified dimension. These results clarify the practical importance of our proposed conditions."/> <meta name="dc.creator" content="Sugiura, Shuhei"/> <meta name="dc.creator" content="Ariizumi, Ryo"/> <meta name="dc.creator" content="Asai, Toru"/> <meta name="dc.creator" content="Azuma, Shun-ichi"/> <meta name="dc.subject" content="Applied mathematics"/> <meta name="dc.subject" content="Computational science"/> <meta name="citation_reference" content="Jaeger, H. The “echo state” approach to analysing and training recurrent neural networks—with an erratum note. 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Reservoir computing is a method for dynamical system approximation that trains the static part of a model but fixes the dynamical part called the reservoir. Hence, reservoir computing has the advantage of training models with a low computational cost. Moreover, fixed reservoirs can be implemented as physical systems. Such reservoirs have attracted attention in terms of computation speed and energy consumption. The universality of a reservoir computing model is its ability to approximate an arbitrary system with arbitrary accuracy. Two sufficient reservoir conditions to make the model universal have been proposed. The first is the combination of fading memory and the separation property. The second is the neighborhood separation property, which we proposed recently. To date, it has been unknown whether a reservoir with a finite-dimensional output can satisfy these conditions. In this study, we prove that no reservoir with a finite-dimensional output satisfies the former condition. By contrast, we propose a single output reservoir that satisfies the latter condition. This implies that, for any dimension, a reservoir making the model universal exists with the output of that specified dimension. These results clarify the practical importance of our proposed conditions."/> <meta property="og:image" content="https://media.springernature.com/full/springer-static/cover-hires/journal/41598"/> <meta name="format-detection" content="telephone=no"> <link rel="apple-touch-icon" sizes="180x180" href=/oscar-static/img/favicons/darwin/apple-touch-icon-92e819bf8a.png> <link rel="icon" type="image/png" sizes="192x192" href=/oscar-static/img/favicons/darwin/android-chrome-192x192-6f081ca7e5.png> <link rel="icon" type="image/png" sizes="32x32" href=/oscar-static/img/favicons/darwin/favicon-32x32-1435da3e82.png> <link rel="icon" type="image/png" sizes="16x16" href=/oscar-static/img/favicons/darwin/favicon-16x16-ed57f42bd2.png> <link rel="shortcut icon" data-test="shortcut-icon" href=/oscar-static/img/favicons/darwin/favicon-c6d59aafac.ico> <meta name="theme-color" content="#e6e6e6">   <link rel="stylesheet" media="print" href=/oscar-static/app-springerlink/css/print-b8af42253b.css> <style> 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Reservoir computing is a method for dynamical system approximation that trains the static part of a model but fixes the dynamical part called the reservoir. Hence, reservoir computing has the advantage of training models with a low computational cost. Moreover, fixed reservoirs can be implemented as physical systems. Such reservoirs have attracted attention in terms of computation speed and energy consumption. The universality of a reservoir computing model is its ability to approximate an arbitrary system with arbitrary accuracy. Two sufficient reservoir conditions to make the model universal have been proposed. The first is the combination of fading memory and the separation property. The second is the neighborhood separation property, which we proposed recently. To date, it has been unknown whether a reservoir with a finite-dimensional output can satisfy these conditions. In this study, we prove that no reservoir with a finite-dimensional output satisfies the former condition. By contrast, we propose a single output reservoir that satisfies the latter condition. This implies that, for any dimension, a reservoir making the model universal exists with the output of that specified dimension. These results clarify the practical importance of our proposed conditions.","datePublished":"2024-04-11T00:00:00Z","dateModified":"2024-04-11T00:00:00Z","pageStart":"1","pageEnd":"10","license":"http://creativecommons.org/licenses/by/4.0/","sameAs":"https://doi.org/10.1038/s41598-024-56742-7","keywords":["Applied mathematics","Computational science","Machine learning","Neural network","Nonlinear dynamical system","Reservoir computing","Science","Humanities and Social Sciences","multidisciplinary"],"image":[],"isPartOf":{"name":"Scientific Reports","issn":["2045-2322"],"volumeNumber":"14","@type":["Periodical","PublicationVolume"]},"publisher":{"name":"Nature Publishing Group UK","logo":{"url":"https://www.springernature.com/app-sn/public/images/logo-springernature.png","@type":"ImageObject"},"@type":"Organization"},"author":[{"name":"Shuhei Sugiura","affiliation":[{"name":"Nagoya University","address":{"name":"Graduate School of Engineering, Nagoya University, Nagoya, 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class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Shuhei-Sugiura-Aff1" data-author-popup="auth-Shuhei-Sugiura-Aff1" data-author-search="Sugiura, Shuhei">Shuhei Sugiura</a><sup class="u-js-hide"><a href="#Aff1">1</a></sup>, </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Ryo-Ariizumi-Aff2" data-author-popup="auth-Ryo-Ariizumi-Aff2" data-author-search="Ariizumi, Ryo" data-corresp-id="c1">Ryo Ariizumi<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><sup class="u-js-hide"><a href="#Aff2">2</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Toru-Asai-Aff1" data-author-popup="auth-Toru-Asai-Aff1" data-author-search="Asai, Toru">Toru Asai</a><sup class="u-js-hide"><a href="#Aff1">1</a></sup> & </li><li class="c-article-author-list__show-more" aria-label="Show all 4 authors for this article" title="Show all 4 authors for this article">…</li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Shun_ichi-Azuma-Aff3" data-author-popup="auth-Shun_ichi-Azuma-Aff3" data-author-search="Azuma, Shun-ichi">Shun-ichi Azuma</a><sup class="u-js-hide"><a href="#Aff3">3</a></sup> </li></ul><button aria-expanded="false" class="c-article-author-list__button"><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-down-medium"></use></svg><span>Show authors</span></button> <div data-test="article-metrics"> <ul class="app-article-metrics-bar u-list-reset"> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-accesses-medium"></use> </svg>830 <span class="app-article-metrics-bar__label">Accesses</span></p> </li> <li class="app-article-metrics-bar__item app-article-metrics-bar__item--metrics"> <p class="app-article-metrics-bar__details"><a href="/article/10.1038/s41598-024-56742-7/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Explore all metrics <svg class="u-icon app-article-metrics-bar__arrow-icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-arrow-right-medium"></use> </svg></a></p> </li> </ul> </div> <div class="u-mt-32"> </div> </header> </div> <div data-article-body="true" data-track-component="article body" class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>In this paper, we prove the existence of a reservoir that has a finite-dimensional output and makes the reservoir computing model universal. Reservoir computing is a method for dynamical system approximation that trains the static part of a model but fixes the dynamical part called the reservoir. Hence, reservoir computing has the advantage of training models with a low computational cost. Moreover, fixed reservoirs can be implemented as physical systems. Such reservoirs have attracted attention in terms of computation speed and energy consumption. The universality of a reservoir computing model is its ability to approximate an arbitrary system with arbitrary accuracy. Two sufficient reservoir conditions to make the model universal have been proposed. The first is the combination of fading memory and the separation property. The second is the neighborhood separation property, which we proposed recently. To date, it has been unknown whether a reservoir with a finite-dimensional output can satisfy these conditions. In this study, we prove that no reservoir with a finite-dimensional output satisfies the former condition. By contrast, we propose a single output reservoir that satisfies the latter condition. This implies that, for any dimension, a reservoir making the model universal exists with the output of that specified dimension. These results clarify the practical importance of our proposed conditions.</p></div></div></section> <div data-test="cobranding-download"> </div> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1007%2Fs11071-024-10178-w/MediaObjects/11071_2024_10178_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s11071-024-10178-w?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1007/s11071-024-10178-w">Polynomial function error stair of reservoir computing and its applications in characterizing the learning capability </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">23 August 2024</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1007%2Fs10884-022-10159-w/MediaObjects/10884_2022_10159_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s10884-022-10159-w?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1007/s10884-022-10159-w">Learning Dynamics by Reservoir Computing (In Memory of Prof. Pavol Brunovský) </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">26 April 2022</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41598-022-20331-3/MediaObjects/41598_2022_20331_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1038/s41598-022-20331-3?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1038/s41598-022-20331-3">Time series reconstructing using calibrated reservoir computing </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">29 September 2022</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; 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An RC model consists of a dynamical system called a reservoir and a static function called a readout. First, the input signal to the RC model is processed in the reservoir. Next, the signal from the reservoir is processed using the readout, and the model output is obtained. The concept of RC is that only the static part of the model is trained, that is, the readout, to make the model behave as desired<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Jaeger, H. The “echo state” approach to analysing and training recurrent neural networks—with an erratum note. German National Research Center for Information Technology GMD Technical Report, 148.34 (2001)." href="#ref-CR1" id="ref-link-section-d117352412e409">1</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="#ref-CR2" id="ref-link-section-d117352412e409_1">2</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Steil, J. J. Backpropagation-decorrelation: Online recurrent learning with O(N) complexity. In 2004 IEEE International Joint Conference on Neural Networks 843–848 (2004)." href="#ref-CR3" id="ref-link-section-d117352412e409_2">3</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Verstraeten, D., Schrauwen, B., D’Haene, M. & Stroobandt, D. An experimental unification of reservoir computing methods. Neural Netw. 20(3), 391–403 (2007)." href="#ref-CR4" id="ref-link-section-d117352412e409_3">4</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="Lukoševičius, M. & Jaeger, H. Reservoir computing approaches to recurrent neural network training. Comput. Sci. Rev. 3(3), 127–149 (2009)." href="/article/10.1038/s41598-024-56742-7#ref-CR5" id="ref-link-section-d117352412e412">5</a></sup>. RC was proposed initially to simplify the training of recurrent neural networks (RNNs)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Williams, R. J. & Zipser, D. A learning algorithm for continually running fully recurrent neural networks. Neural Comput. 1(2), 270–280 (1989)." href="/article/10.1038/s41598-024-56742-7#ref-CR6" id="ref-link-section-d117352412e416">6</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Werbos, P. J. Backpropagation through time: what it does and how to do it. Proc. IEEE 78(10), 1550–1560 (1990)." href="/article/10.1038/s41598-024-56742-7#ref-CR7" id="ref-link-section-d117352412e419">7</a></sup> and is superior in terms of computational cost for training. Generally, a randomly generated RNN is used as the reservoir. However, because the reservoir is fixed, a physical system that is difficult to adjust can also be used as the reservoir. Recently, physical RC, which uses a physical system as the reservoir, has received considerable attention<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Tanaka, G. et al. Recent advances in physical reservoir computing: A review. Neural Netw. 115, 100–123 (2019)." href="#ref-CR8" id="ref-link-section-d117352412e423">8</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Friedman, J. S. Unsupervised learning & reservoir computing leveraging analog spintronic phenomena. IEEE 16th Nanotechnology Materials and Devices Conference 1–2 (2021)." href="#ref-CR9" id="ref-link-section-d117352412e423_1">9</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Stelzer, F., Röhm, A., Lüdge, K. & Yanchuk, S. Performance boost of time-delay reservoir computing by non-resonant clock cycle. Neural Netw. 124, 158–169 (2020)." href="#ref-CR10" id="ref-link-section-d117352412e423_2">10</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Soriano, M. C. et al. Delay-based reservoir computing: Noise effects in a combined analog and digital implementation. IEEE Trans. Neural Netw. Learn. Syst. 26(2), 388–393 (2014)." href="#ref-CR11" id="ref-link-section-d117352412e423_3">11</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Dong, J., Rafayelyan, M., Krzakala, F. & Gigan, S. Optical reservoir computing using multiple light scattering for chaotic systems prediction. IEEE J. Sel. Top. Quantum Electron. 26(1), 1–12 (2020)." href="/article/10.1038/s41598-024-56742-7#ref-CR12" id="ref-link-section-d117352412e426">12</a></sup>. Physical reservoirs are expected to be superior to an RNN implemented on a general-purpose computer in terms of processing speed and energy consumption<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Tanaka, G. et al. Recent advances in physical reservoir computing: A review. Neural Netw. 115, 100–123 (2019)." href="/article/10.1038/s41598-024-56742-7#ref-CR8" id="ref-link-section-d117352412e430">8</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="Nakajima, K. & Fischer, I. Reservoir Computing (Springer, 2021)." href="/article/10.1038/s41598-024-56742-7#ref-CR13" id="ref-link-section-d117352412e433">13</a></sup>.</p><p>We say that an RC model is universal if it can approximate an arbitrary system with arbitrary accuracy by training only the readout. Several studies have been performed on the approximation ability of RC models. Grigoryeva et al.<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Grigoryeva, L. & Ortega, J. P. Echo state networks are universal. Neural Netw. 108, 495–508 (2018)." href="/article/10.1038/s41598-024-56742-7#ref-CR14" id="ref-link-section-d117352412e440">14</a></sup> studied an echo state network (ESN)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Jaeger, H. The “echo state” approach to analysing and training recurrent neural networks—with an erratum note. German National Research Center for Information Technology GMD Technical Report, 148.34 (2001)." href="/article/10.1038/s41598-024-56742-7#ref-CR1" id="ref-link-section-d117352412e444">1</a></sup>, which is a typical RC model composed of an RNN reservoir and linear readout. Gonon et al.<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="Gonon, L. & Ortega, J. P. Reservoir Computing Universality With Stochastic Inputs. IEEE Trans. Neural Netw. Learn. Syst. 31(1), 100–112 (2020)." href="/article/10.1038/s41598-024-56742-7#ref-CR15" id="ref-link-section-d117352412e448">15</a></sup> studied the three classes of RC models for stochastic inputs: one composed of a linear reservoir and polynomial readout, one composed of a state-affine reservoir and linear readout, and an ESN. They showed that each model can approximate any system with any accuracy. However, their results differ from the universality that we deal with in this paper. This is because, for each target system to be approximated, they must train both the readout and the reservoir. Fixing the reservoir regardless of the targets is the concept of RC, and by violating it, advantages such as the hardware implementation of the reservoir and reduction of the computational cost of training are lost.</p><p>We say that a reservoir is universal if it makes the RC model universal. To the best of our knowledge, it is unknown whether a universal reservoir with a finite-dimensional output exists. This is an important problem because a reservoir’s output is finite in practice. The keys to solving this problem are the study of Maass et al.<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e455">2</a></sup> and our recent study<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e459">16</a></sup>. In each of these studies on continuous-time RC, a sufficient condition for a reservoir to be universal was proposed. They use a polynomial readout, evaluate the approximation error using the uniform norm, and assume that the target has fading memory. Fading memory means that if two inputs were close to each other in the recent past, the present outputs are also close. In<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e463">2</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e466">16</a></sup>, an <i>m</i>-output reservoir is represented as a set of <i>m</i> operators, which are maps between input functions and scalar-valued output functions. Therefore, we call a reservoir with a finite-dimensional output a finite reservoir. In<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e477">2</a></sup>, input and output functions of the operators are defined on the bi-infinite-time (BIT) interval <span class="mathjax-tex">${\mathbb {R}}$</span>. Hence, we call these operators BIT operators. Maass et al.<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e499">2</a></sup> showed that a reservoir is universal if it has the separation property and operators in it have fading memory. In<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e503">16</a></sup>, input and output functions of the operators are defined on the right-infinite-time (RIT) interval <span class="mathjax-tex">${\mathbb {R}}_+=\left[ 0,\infty \right)$</span>. Hence, we call these operators RIT operators. In<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e544">16</a></sup>, we showed that a reservoir is universal if it has the neighborhood separation property (NSP) and the operators in it are bounded. However, it remains an open question whether there exists any finite reservoir satisfying those conditions<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e549">2</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e552">16</a></sup>.</p><p>In this paper, we provide two results. First, we show that no finite reservoir satisfies the condition in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e559">2</a></sup>. We derive a contradiction from the assumption that an <i>m</i>-output reservoir satisfies the condition in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e566">2</a></sup>: the separation property and fading memory. BIT operators have a one-to-one correspondence between functionals, which are maps from input signals to <span class="mathjax-tex">${\mathbb {R}}$</span>. Using the functionals, we construct a map from the compact space of input signals to <span class="mathjax-tex">${\mathbb {R}}^m$</span>. The separation property and fading memory mean that the constructed map is injective and continuous, respectively. This leads to the contradiction that the space of input signals and a subset of <span class="mathjax-tex">${\mathbb {R}}^m$</span> are homeomorphic, although they have different dimensions. As the second result, we show that there is a reservoir that has a single output and the NSP, which is the condition in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e641">16</a></sup>. RIT operators also have a one-to-one correspondence between functionals. We show that a single output reservoir with the NSP exists if a functional with a continuous left inverse exists. To obtain such a functional, we use the Hahn–Mazurkiewicz theorem, which provides continuous surjection from <span class="mathjax-tex">$\left[ 0,1\right]$</span> to the space of functional inputs. Assuming the axiom of choice, we can take the right inverse of the surjection, which is the functional that we seek.</p><p>Our contribution in this study is to show that there is no finite reservoir satisfying the condition in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e673">2</a></sup> but there is one satisfying the condition in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e677">16</a></sup>. Through the discussion, we provide an example of a universal reservoir with a single output. The mathematical meaning of our example is that an operator, which is a map between functions, can be approximated by training only the readout, which is a continuous map from <span class="mathjax-tex">$\left[ 0,1\right]$</span> to <span class="mathjax-tex">${\mathbb {R}}$</span>. This is a counter-intuitive and interesting result. The practical meaning of our example is that a reservoir has the possibility of being universal, regardless of the dimension of the output. This is particularly important for using physical reservoirs because they are difficult to adjust.</p><p>The structure of this paper is as follows: first, we show that there is no finite reservoir that satisfies the condition in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e726">2</a></sup>. Second, we show that a universal reservoir with a single output exists that satisfies the condition in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e730">16</a></sup>. Finally, we conclude the paper.</p></div></div></section><section data-title="Reservoir computing represented by bi-infinite-time operators"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2">Reservoir computing represented by bi-infinite-time operators</h2><div class="c-article-section__content" id="Sec2-content"><p>In the study by Maass et al.<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e742">2</a></sup>, which is one of the earliest on RC, they proposed a condition for the reservoir to be universal. We first consider this condition to prove the existence of a universal finite reservoir. Eventually, we conclude that no finite reservoir can satisfy the condition in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e746">2</a></sup>. In the first subsection, we briefly review<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e750">2</a></sup>, and in the second subsection, we describe one of the two main results of the present study.</p><h3 class="c-article__sub-heading" id="Sec3">Condition for universality</h3><p>RC is a method to approximate a dynamical system, a map between functions of time. Let <span class="mathjax-tex">$A\subset {\mathbb {R}}^n$</span> be a compact set of input values and <span class="mathjax-tex">$K>0$</span> be the limit of the speed of input change. We define a set <span class="mathjax-tex">$U^B$</span> of BIT inputs as follows:</p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} U^B = \left\{ u:{\mathbb {R}} \rightarrow A\;\left| \;\forall t_1,t_2 \in {\mathbb {R}},\left\| u{\left( t_1 \right) } - u{\left( t_2 \right) } \right\| \le K \left| {t_1-t_2} \right| \right. \right\} . \end{aligned}$$</span></div><div class="c-article-equation__number"> (1) </div></div><p>Let <span class="mathjax-tex">$Y^B$</span> be a set of output functions from <span class="mathjax-tex">${\mathbb {R}}$</span> to <span class="mathjax-tex">${\mathbb {R}}$</span>. We call operators from <span class="mathjax-tex">$U^B$</span> to <span class="mathjax-tex">$Y^B$</span> “BIT operators” because they are maps between signals defined on the BIT interval <span class="mathjax-tex">${\mathbb {R}}$</span>. For operator <i>F</i> and input signal <i>u</i>, we write the output signal and its value at time <i>t</i> as <i>Fu</i> and <span class="mathjax-tex">$Fu{\left( t\right) }$</span>, respectively. Maass et al.<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e1149">2</a></sup> defined the reservoir, that is, the dynamical part of a RC model, as a set <span class="mathjax-tex">${\mathbb {F}}$</span> of BIT operators. If the cardinal number <span class="mathjax-tex">$m=\left| {\mathbb {F}}\right|$</span> is finite, <span class="mathjax-tex">${\mathbb {F}}$</span> represents a dynamical system that returns <i>m</i> output signals <span class="mathjax-tex">$F_1u,\ldots ,F_mu\in Y^B$</span> for an input signal <span class="mathjax-tex">$u\in U^B$</span>. Hence, we call <i>m</i> the output dimension of <span class="mathjax-tex">${\mathbb {F}}$</span> and call <span class="mathjax-tex">${\mathbb {F}}$</span> an <i>m</i>-output reservoir. Mathematically, <span class="mathjax-tex">${\mathbb {F}}$</span> can even be an uncountable set: for example, the reservoir representing waves on a liquid surface<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Fernando, C. & Sojakka, S. Pattern recognition in a bucket. In European Conference on Artificial Life 588–597 (Springer, 2003)." href="/article/10.1038/s41598-024-56742-7#ref-CR17" id="ref-link-section-d117352412e1363">17</a></sup> has the cardinality of continuum.</p><p>The RC model <span class="mathjax-tex">${\hat{F}}:U^B\rightarrow Y^B$</span> is defined as follows:</p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\hat{F}}u{\left( t\right) }=p{\left( F_1u{\left( t\right) },\dots ,F_iu{\left( t\right) }\right) }\quad \left( u\in U^B,t\in {\mathbb {R}}\right) , \end{aligned}$$</span></div><div class="c-article-equation__number"> (2) </div></div><p>where <span class="mathjax-tex">$i\in {\mathbb {N}}$</span> and <span class="mathjax-tex">$F_1,\dots ,F_i\in {\mathbb {F}}$</span>. The function <span class="mathjax-tex">$p:{\mathbb {R}}^i\rightarrow {\mathbb {R}}$</span> is a polynomial called a readout. If <i>m</i> is finite, we can set <span class="mathjax-tex">$i=m$</span> and <span class="mathjax-tex">$\left\{ F_1,\dots ,F_i\right\} ={\mathbb {F}}$</span>. In this case, the RC model is trained only by turning the readout <i>p</i>. However, if <i>m</i> is infinite, we must also select a finite number of operators <span class="mathjax-tex">$F_1,\dots ,F_i\in {\mathbb {F}}$</span>. This operation effectively means the training of the reservoir as well as the readout. For instance, in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Boyd, S. & Chua, L. O. Fading memory and the problem of approximating nonlinear operators with Volterra series. IEEE Trans. Circuits Syst. 32(11), 1150–1161 (1985)." href="/article/10.1038/s41598-024-56742-7#ref-CR18" id="ref-link-section-d117352412e1766">18</a></sup>, they considered a reservoir composed of all operators defined by a stable linear system. In practice, selecting the operators from such a reservoir means the turning of some parameters of a linear system. Therefore, a finite reservoir is required to achieve the low computational cost of training, which is the advantage of RC. In the remainder of this paper, we assume that <i>m</i> is finite unless otherwise specified.</p><p>The RC model and its reservoir are said to be universal if the model can approximate an arbitrary dynamical system with arbitrary accuracy. We formulate the universality of a reservoir represented by BIT operators as follows:</p> <h3 class="c-article__sub-heading" id="FPar1">Definition 1</h3> <p>Let reservoir <span class="mathjax-tex">${\mathbb {F}}$</span> be a set of BIT operators <span class="mathjax-tex">$F_1,\dots ,F_m$</span> and let <span class="mathjax-tex">${\mathbb {F}}^*$</span> be another set of BIT operators. Reservoir <span class="mathjax-tex">${\mathbb {F}}$</span> is said to be universal for uniform approximations in <span class="mathjax-tex">${\mathbb {F}}^*$</span> if, for any operator <span class="mathjax-tex">$F^*\in {\mathbb {F}}^*$</span> and <span class="mathjax-tex">$\varepsilon >0$</span>, a polynomial <span class="mathjax-tex">$p:{{\mathbb {R}}}^m \rightarrow {\mathbb {R}}$</span> exists that satisfies</p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \forall u\in U^B,\forall t\in {\mathbb {R}},\left| F^*u{\left( t \right) }-p{\left( F_1u{\left( t \right) },\ldots ,F_mu{\left( t \right) } \right) } \right| <\varepsilon . \end{aligned}$$</span></div><div class="c-article-equation__number"> (3) </div></div> <p>Definition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar1">1</a> means that, if the reservoir <span class="mathjax-tex">${\mathbb {F}}$</span> is universal, a polynomial of operators <span class="mathjax-tex">$F_1,\dots ,F_m$</span> can approximate any target in <span class="mathjax-tex">${\mathbb {F}}^*$</span> with any accuracy.</p><p>To guarantee the universality of RC, Maass et al.<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e2229">2</a></sup> considered BIT operators that have two properties called time invariance and fading memory. An operator <span class="mathjax-tex">$F:U^B\rightarrow Y^B$</span> is said to be time-invariant if the following holds:</p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \forall t\in {\mathbb {R}}, \forall u_1,u_2\in U^B,\left[ \left( \forall \tau \in {\mathbb {R}}, u_2{\left( \tau \right) }=u_1{\left( \tau -t\right) }\right) \Rightarrow \left( \forall \tau \in {\mathbb {R}}, Fu_2{\left( \tau \right) }=Fu_1{\left( \tau -t\right) }\right) \right] . \end{aligned}$$</span></div><div class="c-article-equation__number"> (4) </div></div><p>The property time invariance means that a temporal shift of an input also shifts the output.</p><p>Fading memory is defined as follows:</p> <h3 class="c-article__sub-heading" id="FPar2">Definition 2</h3> <p>A BIT operator <span class="mathjax-tex">$F:U^B\rightarrow Y^B$</span> is said to have fading memory if the following holds:</p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \forall u_1\in U^B,\forall \varepsilon>0,\exists \delta>0,\exists T>0,\forall u_2\in U^B,\left[ \max _{\tau \in \left[ -T,0\right] }\left\| u_1{\left( \tau \right) }-u_2{\left( \tau \right) }\right\|<\delta \Rightarrow \left| Fu_1{\left( 0\right) }-Fu_2{\left( 0\right) }\right| <\varepsilon \right] . \end{aligned}$$</span></div><div class="c-article-equation__number"> (5) </div></div> <p>Fading memory means that the output value strongly depends on the recent past input but weakly depends on the distant past. Fading memory also means independence from the future, which we define as causality later in the paper.</p><p>According to Maass et al.<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e2714">2</a></sup>, the following condition is essential for a reservoir to be universal.</p> <h3 class="c-article__sub-heading" id="FPar3">Definition 3</h3> <p>Let reservoir <span class="mathjax-tex">${\mathbb {F}}$</span> be a set of BIT operators. Reservoir <span class="mathjax-tex">${\mathbb {F}}$</span> is said to have the separation property if <span class="mathjax-tex">${\mathbb {F}}$</span> satisfies the following:</p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \forall u_1,u_2\in U^B,\exists F\in {\mathbb {F}},\left[ \left( \exists \tau \le 0,u_1{\left( \tau \right) }\ne u_2{\left( \tau \right) }\right) \Rightarrow Fu_1{\left( 0\right) }\ne Fu_2{\left( 0\right) }\right] . \end{aligned}$$</span></div><div class="c-article-equation__number"> (6) </div></div> <p>The separation property means that the reservoir provides different outputs to different inputs. Suppose that the reservoir does not have the separation property, that is, the reservoir returns the same output to two different inputs. Then, the RC model cannot approximate a target that returns different outputs to those inputs. Hence, the separation property is necessary to achieve universality. Note that Eqs. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ5">5</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ6">6</a>) are the conditions for the output at time 0 because time invariance is assumed. Under time invariance, the same holds at times other than 0.</p><p>The result in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e2945">2</a></sup> is described as follows:</p> <h3 class="c-article__sub-heading" id="FPar4">Theorem 1</h3> <p>(Maass and Natschl<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e2956">2</a></sup>) Let <span class="mathjax-tex">${\mathbb {F}}^*$</span> be the set of time-invariant BIT operators with fading memory. Suppose that reservoir <span class="mathjax-tex">${\mathbb {F}}\subset {\mathbb {F}}^*$</span> has the separation property. Then, reservoir <span class="mathjax-tex">${\mathbb {F}}$</span> is universal for uniform approximations in <span class="mathjax-tex">${\mathbb {F}}^*$</span>.</p> <p>To summarize the result in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e3068">2</a></sup>, a reservoir with the separation property that contains only time-invariant operators with fading memory is universal. If <i>m</i> can be infinite, a reservoir that satisfies the condition of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar4">1</a> exists. For example, the reservoir composed of linear systems<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Boyd, S. & Chua, L. O. Fading memory and the problem of approximating nonlinear operators with Volterra series. IEEE Trans. Circuits Syst. 32(11), 1150–1161 (1985)." href="/article/10.1038/s41598-024-56742-7#ref-CR18" id="ref-link-section-d117352412e3078">18</a></sup>, which we mentioned before, satisfies the condition.</p><h3 class="c-article__sub-heading" id="Sec4">No finite reservoir satisfies the condition of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar4">1</a> </h3><p>We show that any finite reservoir does not satisfy the condition of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar4">1</a>. To achieve this, we define a functional corresponding to a time-invariant and causal BIT operator, with which fading memory and the separation property are expressed more simply. The causality of a BIT operator is defined as follows:</p> <h3 class="c-article__sub-heading" id="FPar5">Definition 4</h3> <p>BIT operator <span class="mathjax-tex">$F:U^B\rightarrow Y^B$</span> is said to be causal if <i>F</i> satisfies</p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \forall u_1,u_2\in U^B,\forall t\in {\mathbb {R}},\left[ \left( \forall \tau \le t, u_1{\left( \tau \right) }=u_2{\left( \tau \right) }\right) \Rightarrow Fu_1{\left( t\right) }=Fu_2{\left( t\right) }\right] . \end{aligned}$$</span></div><div class="c-article-equation__number"> (7) </div></div> <p>A BIT operator with fading memory is causal. With causality, the output value <span class="mathjax-tex">$Fu{\left( t\right) }$</span> depends only on the input values of <i>u</i> on <span class="mathjax-tex">$\left( -\infty ,t\right]$</span>. Moreover, with time invariance, this relation does not depend on <i>t</i>. Hence, a causal and time-invariant operator is considered as a relation between input signals on <span class="mathjax-tex">${\mathbb {R}}_-=\left( -\infty ,0\right]$</span> and <span class="mathjax-tex">${\mathbb {R}}$</span>. The corresponding functional is defined by such a relation. For the compact set <span class="mathjax-tex">$A\subset {\mathbb {R}}^n$</span> and <span class="mathjax-tex">$K>0$</span> in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ1">1</a>), we define the domain <i>V</i> of functionals as follows:</p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} V = \left\{ v:{\mathbb {R}}_- \rightarrow A\;\left| \;\forall t_1,t_2 \le 0,\left\| v{\left( t_1 \right) } - v{\left( t_2 \right) } \right\| \le K \left| {t_1-t_2} \right| \right. \right\} . \end{aligned}$$</span></div><div class="c-article-equation__number"> (8) </div></div><p>The correspondence between a BIT operator and a functional is defined as follows:</p> <h3 class="c-article__sub-heading" id="FPar6">Definition 5</h3> <p>A causal and time-invariant BIT operator <span class="mathjax-tex">$F:U^B\rightarrow Y^B$</span> and a functional <span class="mathjax-tex">$f:V\rightarrow {\mathbb {R}}$</span> are said to correspond to each other if the following holds:</p><div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \forall u\in U^B,\forall v\in V, \left[ \left( \forall \tau \le 0,u{\left( \tau \right) }=v{\left( \tau \right) }\right) \Rightarrow Fu{\left( 0\right) }=f{\left( v\right) }\right] . \end{aligned}$$</span></div><div class="c-article-equation__number"> (9) </div></div> <p>This correspondence is one-to-one<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Boyd, S. & Chua, L. O. Fading memory and the problem of approximating nonlinear operators with Volterra series. IEEE Trans. Circuits Syst. 32(11), 1150–1161 (1985)." href="/article/10.1038/s41598-024-56742-7#ref-CR18" id="ref-link-section-d117352412e3825">18</a></sup> and we can express the conditions of operators using their corresponding functionals.</p><p>We define a norm on <i>V</i> and a metric derived from it. Let <span class="mathjax-tex">$w:{\mathbb {R}}_+\rightarrow \left( 0,1 \right]$</span> be a non-increasing function that satisfies <span class="mathjax-tex">$\lim _{t\rightarrow \infty }w{\left( t\right) }=0$</span>. We define a weighted norm <span class="mathjax-tex">$\left\| v\right\| _w$</span> of <span class="mathjax-tex">$v\in V$</span> as follows:</p><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left\| v\right\| _w=\sup _{\tau \le 0}\left\| v{\left( \tau \right) }\right\| w{\left( -\tau \right) }. \end{aligned}$$</span></div><div class="c-article-equation__number"> (10) </div></div><p>Using this norm, we define a metric <span class="mathjax-tex">$d:V\times V\rightarrow {\mathbb {R}}_+$</span> on <i>V</i> as follows:</p><div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} d{\left( v_1,v_2\right) }=\left\| v_1-v_2\right\| _w. \end{aligned}$$</span></div><div class="c-article-equation__number"> (11) </div></div><p>Hereafter, we use <i>d</i> as a metric on <i>V</i>. Fading memory and the separation property are expressed by a functional as follows:</p> <h3 class="c-article__sub-heading" id="FPar7">Proposition 1</h3> <p>A causal and time-invariant operator <span class="mathjax-tex">$F:U^B\rightarrow Y^B$</span> has fading memory if and only if the functional <span class="mathjax-tex">$f:V\rightarrow {\mathbb {R}}$</span> corresponding to <i>F</i> is continuous.</p> <h3 class="c-article__sub-heading" id="FPar8">Proposition 2</h3> <p>Let <span class="mathjax-tex">$F_1,\ldots ,F_m:U^B\rightarrow Y^B$</span> be causal and time-invariant operators and functionals <span class="mathjax-tex">$f_1,\ldots ,f_m:V\rightarrow {\mathbb {R}}$</span> correspond to <span class="mathjax-tex">$F_1,\ldots ,F_m$</span>, respectively. Then, reservoir <span class="mathjax-tex">${\mathbb {F}}=\left\{ F_1,\ldots ,F_m\right\}$</span> has the separation property if and only if the following map <span class="mathjax-tex">$\left( f_1,\ldots ,f_m\right) :V\rightarrow {\mathbb {R}}^m$</span> is injective:</p><div id="Equ12" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left( f_1,\ldots ,f_m\right) {\left( v\right) }=\left( f_1{\left( v\right) },\ldots ,f_m{\left( v\right) }\right) \quad \left( v\in V\right) . \end{aligned}$$</span></div><div class="c-article-equation__number"> (12) </div></div> <p>We provide the proofs in the supplementary material. Using Propositions <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar7">1</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar8">2</a>, we obtain the following theorem, which provides the conclusion of this section.</p> <h3 class="c-article__sub-heading" id="FPar9">Theorem 2</h3> <p>Suppose that the input range <span class="mathjax-tex">$A\subset {\mathbb {R}}^n$</span> in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ1">1</a>) includes distinct <span class="mathjax-tex">$a_1$</span> and <span class="mathjax-tex">$a_2$</span> that satisfy</p><div id="Equ13" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \forall \alpha \in \left[ 0,1\right] ,a_1+\alpha \left( a_2-a_1\right) \in A \end{aligned}.$$</span></div><div class="c-article-equation__number"> (13) </div></div><p>Then, no natural number <i>m</i> and time-invariant operators <span class="mathjax-tex">$F_1,\ldots ,F_m:U^B\rightarrow Y^B$</span> with fading memory exist such that reservoir <span class="mathjax-tex">${\mathbb {F}}=\left\{ F_1,\ldots ,F_m\right\}$</span> has the separation property.</p> <h3 class="c-article__sub-heading" id="FPar10">Proof of Theorem 2</h3> <p>We prove the theorem by contradiction. Suppose that natural number <i>m</i> and time-invariant operators <span class="mathjax-tex">$F_1,\ldots ,F_m:U^B\rightarrow Y^B$</span> with fading memory exist such that reservoir <span class="mathjax-tex">${\mathbb {F}}=\left\{ F_1,\ldots ,F_m\right\}$</span> has the separation property. Let functionals <span class="mathjax-tex">$f_1,\ldots ,f_m:V\rightarrow {\mathbb {R}}$</span> correspond to <span class="mathjax-tex">$F_1,\ldots ,F_m$</span>, respectively. Then, from Propositions <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar7">1</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar8">2</a>, the map <span class="mathjax-tex">$\left( f_1,\ldots ,f_m\right) :V\rightarrow {\mathbb {R}}^m$</span> is a continuous injection. An injection can be considered as a bijection onto its image, and a continuous bijection from a compact domain to a Hausdorff space is a homeomorphism. Because the set <i>V</i> is compact<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e5196">16</a></sup>, map <span class="mathjax-tex">$\left( f_1,\ldots ,f_m\right)$</span> is a topological embedding, that is, a homeomorphism onto its image. We use the following lemma. <span class="mathjax-tex">$\square$</span></p> <h3 class="c-article__sub-heading" id="FPar11">Lemma 1</h3> <p>A topological embedding <span class="mathjax-tex">$g:\left[ 0,1\right] ^{m+1}\rightarrow V$</span> exists.</p> <p>We provide the proof after the Proof of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar9">2</a>. From Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar11">1</a>, the composition <span class="mathjax-tex">$\left( f_1,\ldots ,f_m\right) \circ g:\left[ 0,1\right] ^{m+1}\rightarrow {\mathbb {R}}^m$</span> is also a topological embedding, that is, <span class="mathjax-tex">$\left[ 0,1\right] ^{m+1}$</span> and <span class="mathjax-tex">$\left( f_1,\ldots ,f_m\right) \circ g\left( \left[ 0,1\right] ^{m+1}\right) \subset {\mathbb {R}}^m$</span> are homeomorphic. We call the small inductive dimension simply a dimension. Dimensions have the following two properties (see pages 3–4 of reference<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Engelking, R. Dimension Theory (North-Holland Publishing Company, 1978)." href="/article/10.1038/s41598-024-56742-7#ref-CR19" id="ref-link-section-d117352412e5525">19</a></sup>): first, two homeomorphic topological spaces have the same dimension. Second, a topological space has a dimension equal to its subspace or larger. Therefore, we have</p><div id="Equ14" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} m+1={\textrm{ind}}\;\left[ 0,1\right] ^{m+1}={\textrm{ind}}\;\left( f_1,\ldots ,f_m\right) \circ g\left( \left[ 0,1\right] ^{m+1}\right) \le {\textrm{ind}}\;{\mathbb {R}}^m=m, \end{aligned}$$</span></div><div class="c-article-equation__number"> (14) </div></div><p>where ind<span class="mathjax-tex">$\left( \cdot \right)$</span> is the dimension. Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ14">14</a>) is a contradiction, which proves Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar9">2</a>. <span class="mathjax-tex">$\square$</span></p><p>As shown in Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar9">2</a>, any finite reservoir does not satisfy the condition of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar4">1</a>. Therefore, Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar4">1</a> cannot support the possibility that a universal finite reservoir exists. We prove Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar11">1</a> as follows:</p> <h3 class="c-article__sub-heading" id="FPar12">Proof of Lemma 1</h3> <p>From the assumption, distinct <span class="mathjax-tex">$a_1$</span>, <span class="mathjax-tex">$a_2\in A$</span> exist that satisfy <span class="mathjax-tex">$a_1+\alpha \left( a_2-a_1\right) \in A$</span> for any <span class="mathjax-tex">$\alpha \in \left[ 0,1\right]$</span>. Let <span class="mathjax-tex">$T=\frac{1}{K}\left\| a_1-a_2\right\|$</span>. For <span class="mathjax-tex">$c\in \left[ 0,1\right] ^{m+1}$</span>, we define a continuous piecewise linear function <span class="mathjax-tex">$\alpha _c:{\mathbb {R}}_-\rightarrow \left[ 0,1\right]$</span> as follows:</p><div id="Equ15" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \alpha _c{\left( t\right) }=c_i+\frac{1}{T}\left( -iT-t\right) \left( c_{i+1}-c_i\right) \quad \left( -\left( i+1\right) T<t\le -iT\right) \end{aligned}.$$</span></div><div class="c-article-equation__number"> (15) </div></div><p>where <span class="mathjax-tex">$i\in \left\{ 0\right\} \cup {\mathbb {N}}$</span>, <span class="mathjax-tex">$\left( c_1,\ldots ,c_{m+1}\right) =c$</span>, and <span class="mathjax-tex">$c_0,c_{m+2},c_{m+3},\ldots$</span> are zeros. The function <span class="mathjax-tex">$\alpha _c$</span> satisfies <span class="mathjax-tex">$\alpha _c{\left( -iT\right) }=c_i$</span>. Because the Lipschitz constant of <span class="mathjax-tex">$\alpha _c$</span> is <span class="mathjax-tex">$\frac{K}{\left\| a_1-a_2\right\| }$</span> or less, we can define a continuous function <span class="mathjax-tex">$g:\left[ 0,1\right] ^{m+1}\rightarrow V$</span> as follows:</p><div id="Equ16" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} g:c\mapsto v\quad \left( c\in \left[ 0,1\right] ^{m+1}\right) ,\quad v{\left( t\right) }=a_1+\alpha _c{\left( t\right) }\left( a_2-a_1\right) \quad \left( t\le 0\right) \end{aligned}.$$</span></div><div class="c-article-equation__number"> (16) </div></div><p>Because <span class="mathjax-tex">$v=g{\left( c\right) }$</span> satisfies <span class="mathjax-tex">$v{\left( -iT\right) }=a_1+c_i\left( a_2-a_1\right)$</span> for any <span class="mathjax-tex">$c=\left( c_1,\ldots ,c_{m+1}\right) \in \left[ 0,1\right] ^{m+1}$</span> and <span class="mathjax-tex">$i\in \left\{ 1,\ldots ,m+1\right\}$</span>, <i>g</i> is injective. Therefore, <i>g</i> is a continuous bijection from its compact domain <span class="mathjax-tex">$\left[ 0,1\right] ^{m+1}$</span> to its image, that is, <i>g</i> is a topological embedding, which proves Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar11">1</a>. <span class="mathjax-tex">$\square$</span></p> </div></div></section><section data-title="Reservoir computing represented by right-infinite-time operators"><div class="c-article-section" id="Sec5-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec5">Reservoir computing represented by right-infinite-time operators</h2><div class="c-article-section__content" id="Sec5-content"><p>To obtain the output value of a BIT operator, we have to consider an input signal that continues from the infinite past. In our recent study<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e6907">16</a></sup>, we avoided this impracticality by defining input and output signals for positive time and proposed another condition for the reservoir to be universal. Using this condition, we prove the existence of a universal finite reservoir. In the first subsection, we briefly explain our previous results<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e6911">16</a></sup>. In the second subsection, we prove the main result of the present study.</p><h3 class="c-article__sub-heading" id="Sec6">Condition for universality</h3><p>For compact set <span class="mathjax-tex">$A\subset {\mathbb {R}}^n$</span> and <span class="mathjax-tex">$K>0$</span> in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ1">1</a>), we define a set <span class="mathjax-tex">$U^R$</span> of RIT inputs as follows:</p><div id="Equ17" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{aligned} U^R = \left\{ u:{\mathbb {R}}_+ \rightarrow A\;\left| \;\forall t_1,t_2\ge 0,\left\| u{\left( t_1 \right) } - u{\left( t_2 \right) } \right\| \le K \left| {t_1-t_2} \right| \right. \right\} . \end{aligned} \end{aligned}$$</span></div><div class="c-article-equation__number"> (17) </div></div><p>Let <span class="mathjax-tex">$Y^R$</span> be the set of output functions from <span class="mathjax-tex">${\mathbb {R}}_+$</span> to <span class="mathjax-tex">${\mathbb {R}}$</span>. We call operators from <span class="mathjax-tex">$U^R$</span> to <span class="mathjax-tex">$Y^R$</span> “RIT operators” because they are maps between signals defined on the RIT interval <span class="mathjax-tex">${\mathbb {R}}_+$</span>. In<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e7300">16</a></sup>, the reservoir is defined as a set <span class="mathjax-tex">${\mathbb {F}}$</span> of RIT operators. The RC model <span class="mathjax-tex">${\hat{F}}:U^R\rightarrow Y^R$</span> is defined as follows:</p><div id="Equ18" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\hat{F}}u{\left( t\right) }=p{\left( F_1u{\left( t\right) },\dots ,{F_m}u{\left( t\right) }\right) }\quad \left( u\in U^R,t\ge 0\right) , \end{aligned}$$</span></div><div class="c-article-equation__number"> (18) </div></div><p>where <span class="mathjax-tex">$\left\{ F_1,\dots ,F_m\right\} = {\mathbb {F}}$</span>. The polynomial <span class="mathjax-tex">$p:{{\mathbb {R}}^m}\rightarrow {\mathbb {R}}$</span> is the readout.</p><p>In the RIT operator case, universality is defined as follows:</p> <h3 class="c-article__sub-heading" id="FPar13">Definition 6</h3> <p>Let reservoir <span class="mathjax-tex">${\mathbb {F}}$</span> be a set of RIT operators <span class="mathjax-tex">$F_1,\ldots ,F_m$</span> and let <span class="mathjax-tex">${\mathbb {F}}^*$</span> be another set of RIT operators. Reservoir <span class="mathjax-tex">${\mathbb {F}}$</span> is said to be universal for uniform approximations in <span class="mathjax-tex">${\mathbb {F}}^*$</span> if, for any operator <span class="mathjax-tex">$F^*\in {\mathbb {F}}^*$</span> and <span class="mathjax-tex">$\varepsilon >0$</span>, a polynomial <span class="mathjax-tex">$p:{{\mathbb {R}}}^m \rightarrow {\mathbb {R}}$</span> exists that satisfies</p><div id="Equ19" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \forall u\in U^R,\forall t\ge 0,\left| F^*u{\left( t \right) }-p{\left( F_1u{\left( t \right) },\ldots ,F_mu{\left( t \right) } \right) } \right| <\varepsilon . \end{aligned}$$</span></div><div class="c-article-equation__number"> (19) </div></div> <p>RIT operators also correspond to functionals, which are maps to real numbers. We require these corresponding functionals to describe the result in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e7935">16</a></sup>. To correspond to a functional, an RIT operator must be causal as follows:</p> <h3 class="c-article__sub-heading" id="FPar14">Definition 7</h3> <p>An RIT operator <span class="mathjax-tex">$F:U^R\rightarrow Y^R$</span> is said to be causal if <i>F</i> satisfies</p><div id="Equ20" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \forall u_1,u_2\in U^R,\forall t\ge 0,\left[ \left( \forall \tau \in \left[ 0,t\right] , u_1{\left( \tau \right) }=u_2{\left( \tau \right) }\right) \Rightarrow Fu_1{\left( t\right) }=Fu_2{\left( t\right) }\right] . \end{aligned}$$</span></div><div class="c-article-equation__number"> (20) </div></div> <p>With causality, the output value <span class="mathjax-tex">$Fu{\left( t\right) }$</span> depends only on the input values of <i>u</i> on <span class="mathjax-tex">$\left[ 0,t\right]$</span>. Hence, a causal RIT operator is considered as a relation between input signals of various lengths and <span class="mathjax-tex">${\mathbb {R}}$</span>. The corresponding functional is defined by such a relation. For the set <i>V</i> in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ8">8</a>), we define the domain <span class="mathjax-tex">$V^{\textrm{res}}$</span> of functionals as follows:</p><div id="Equ21" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} V^{\textrm{res}} =\left\{ v^{\left[ t \right] }\;\left| \; v \in V, t \ge 0 \right. \right\} , \end{aligned}$$</span></div><div class="c-article-equation__number"> (21) </div></div><p>where <span class="mathjax-tex">$v^{\left[ t \right] }$</span> is the restriction of <i>v</i> to <span class="mathjax-tex">$\left[ -t,0\right]$</span>. Let <span class="mathjax-tex">$\lambda :V^{\textrm{res}}\rightarrow {\mathbb {R}}_+$</span> be a map defined by <span class="mathjax-tex">$v^{\left[ t \right] }\mapsto t$</span>. For example, <span class="mathjax-tex">$\lambda {\left( v\right) }=1$</span> for <span class="mathjax-tex">$v:{\left[ -1,0\right] }\rightarrow A$</span>. The correspondence between an RIT operator and a functional is defined as follows:</p> <h3 class="c-article__sub-heading" id="FPar15">Definition 8</h3> <p>A causal RIT operator <span class="mathjax-tex">$F:U^R\rightarrow Y^R$</span> and a functional <span class="mathjax-tex">$f:V^{\textrm{res}}\rightarrow {\mathbb {R}}$</span> are said to correspond to each other if the following holds:</p><div id="Equ22" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \forall u\in U^R,\forall v\in V^{\textrm{res}},\left[ \left( \forall \tau \in \left[ -t,0\right] ,u{\left( t+\tau \right) }=v{\left( \tau \right) }\right) \Rightarrow Fu{\left( t\right) }=f{\left( v\right) }\right] , \end{aligned}$$</span></div><div class="c-article-equation__number"> (22) </div></div><p>where <span class="mathjax-tex">$t=\lambda {\left( v\right) }$</span>.</p> <p>Similar to the BIT operator case, this correspondence is one-to-one<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e8772">16</a></sup>.</p><p>We need a metric on <span class="mathjax-tex">$V^{\textrm{res}}$</span> to explain the condition of reservoirs for universality proposed in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e8801">16</a></sup>. Using <span class="mathjax-tex">$w:{\mathbb {R}}_+\rightarrow \left( 0,1 \right]$</span> in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ10">10</a>), we define the weighted norm <span class="mathjax-tex">$\left\| v \right\| _w$</span> of <span class="mathjax-tex">$v:\left[ -t,0 \right] \rightarrow {\mathbb {R}}^n$</span> as</p><div id="Equ23" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left\| v \right\| _w = \sup _{\tau \in \left[ {-t,0} \right] } \left\| v{\left( \tau \right) }\right\| w{\left( -\tau \right) }. \end{aligned}$$</span></div><div class="c-article-equation__number"> (23) </div></div><p>Let <span class="mathjax-tex">$\theta :{\mathbb {R}}_+\rightarrow {\mathbb {R}}_+$</span> be a strictly increasing, bounded, and continuous function. For <i>w</i> and <span class="mathjax-tex">$\theta$</span>, we define the distance <span class="mathjax-tex">$d{\left( {v_1},{v_2} \right) }$</span> between <span class="mathjax-tex">$v_1$</span>, <span class="mathjax-tex">$v_2\in V^{\textrm{res}}$</span> as</p><div id="Equ24" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} d{\left( {v_1},{v_2} \right) }=\left\| v_1^{\left[ t_{\min }\right] }-v_2^{\left[ t_{\min }\right] } \right\| _w+\left| \theta {\left( t_1 \right) }-\theta {\left( t_2 \right) }\right| , \end{aligned}$$</span></div><div class="c-article-equation__number"> (24) </div></div><p>where <span class="mathjax-tex">$t_1=\lambda {\left( v_1\right) }$</span>, <span class="mathjax-tex">$t_2=\lambda {\left( v_2\right) }$</span>, and <span class="mathjax-tex">$t_{\min }=\min \left\{ t_1,t_2 \right\}$</span>. See Assumption 1 in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e9437">16</a></sup> for detailed conditions for the map <span class="mathjax-tex">$d:V^{\textrm{res}}\times V^{\textrm{res}}\rightarrow {\mathbb {R}}_+$</span> to be a metric. We extend <i>d</i> onto <span class="mathjax-tex">$V^{\textrm{res}}\cup V$</span> by defining <span class="mathjax-tex">$\lambda {\left( v\right) }=\infty$</span>, <span class="mathjax-tex">$v^{\left[ \infty \right] }=v$</span>, and <span class="mathjax-tex">$\theta {\left( \infty \right) }=\lim _{t\rightarrow \infty }\theta {\left( t\right) }$</span> for any <span class="mathjax-tex">$v\in V$</span>. With this extension, the metrics in Eqs. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ11">11</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ24">24</a>) are equivalent on <i>V</i>. Hereafter, we use <i>d</i> as a metric on <span class="mathjax-tex">$V^{\textrm{res}}\cup V$</span>. Any <span class="mathjax-tex">$v\in V$</span> is an accumulation point of <span class="mathjax-tex">$V^{\textrm{res}}$</span> because <span class="mathjax-tex">$v^{\left[ t\right] }\in V^{\textrm{res}}$</span> converges to <i>v</i> as <span class="mathjax-tex">$t\rightarrow \infty$</span>. As shown in Proposition 5 in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e9814">16</a></sup>, <span class="mathjax-tex">$V^{\textrm{res}}\cup V$</span> is compact. Therefore, <span class="mathjax-tex">$V^{\textrm{res}}\cup V=\overline{V^{\textrm{res}}}$</span> holds, where <span class="mathjax-tex">$\overline{\;\cdot \;}$</span> means closure. In<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e9919">16</a></sup>, the following condition is proposed for a reservoir to be universal.</p> <h3 class="c-article__sub-heading" id="FPar16">Definition 9</h3> <p>Let <span class="mathjax-tex">${\mathbb {F}}$</span> be a set of causal RIT operators <span class="mathjax-tex">$F_1,\ldots ,F_m$</span> and let <span class="mathjax-tex">$f_1,\ldots ,f_m:V^{\textrm{res}}\rightarrow {\mathbb {R}}$</span> correspond to each operator in <span class="mathjax-tex">${\mathbb {F}}$</span>. The set <span class="mathjax-tex">${\mathbb {F}}$</span> is said to have the neighborhood separation property (NSP) if the following holds for any distinct <span class="mathjax-tex">$v_1$</span>, <span class="mathjax-tex">$v_2\in \overline{V^{\textrm{res}}}$</span>:</p><div id="Equ25" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \exists \delta >0,\overline{\left( f_1,\ldots ,f_m\right) {\left( N_\delta {\left( v_1 \right) }\cap V^{\textrm{res}} \right) }}\cap \overline{\left( f_1,\ldots ,f_m\right) {\left( N_\delta {\left( v_2 \right) }\cap V^{\textrm{res}}\right) }}=\emptyset , \end{aligned}$$</span></div><div class="c-article-equation__number"> (25) </div></div><p>where <span class="mathjax-tex">$N_\delta {\left( v \right) }$</span> is a <span class="mathjax-tex">$\delta$</span>-neighborhood of <span class="mathjax-tex">$v\in \overline{V^{\textrm{res}}}$</span> defined as follows:</p><div id="Equ26" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} N_\delta {\left( v \right) }=\left\{ v'\in \overline{V^{\textrm{res}}}\left| d{\left( v',v\right) }<\delta \right. \right\} . \end{aligned}$$</span></div><div class="c-article-equation__number"> (26) </div></div> <p>The NSP guarantees that the images of neighborhoods of distinct points are disjoint. Hence, we can say that the NSP is the “strong injectivity” of the map <span class="mathjax-tex">$\left( f_1,\ldots ,f_m\right)$</span>.</p><p>In<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e10532">16</a></sup>, target operator <span class="mathjax-tex">$F^*:U^R\rightarrow Y^R$</span> is assumed to correspond to a uniformly continuous functional. As shown in Proposition 8 in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e10580">16</a></sup>, this assumption gives <span class="mathjax-tex">$F^*$</span> three properties. The first is the following:</p><div id="Equ27" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \forall \varepsilon>0,\forall u_1\in U^R,\forall t\ge 0,\exists \delta >0,\forall u_2\in U^R,\left[ \max _{\tau \in \left[ 0,t\right] }\left\| u_1{\left( \tau \right) }-u_2{\left( \tau \right) }\right\|<\delta \Rightarrow \left| F^*u_1{\left( t\right) }-F^*u_2{\left( t\right) }\right| <\varepsilon \right] . \end{aligned}$$</span></div><div class="c-article-equation__number"> (27) </div></div><p>Roughly speaking, this is a “continuity” of the output value with respect to past inputs. Hence, <span class="mathjax-tex">$F^*$</span> is also causal. The second property is the equicontinuity of output signals, that is,</p><div id="Equ28" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \forall \varepsilon>0,\forall t\ge 0,\exists \delta >0,\forall y\in F^*{\left( U\right) },\forall t'\ge 0,\left[ \left| t-t'\right|<\delta \Rightarrow \left| y{\left( t\right) }-y{\left( t'\right) }\right| <\varepsilon \right] . \end{aligned}$$</span></div><div class="c-article-equation__number"> (28) </div></div><p>The third property is fading memory of an RIT operator<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e11003">16</a></sup>, which means that older inputs have less influence on the present output. At first glance, the uniform continuity of the target’s corresponding functional is a stricter condition than that in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e11007">2</a></sup>. However, in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e11011">2</a></sup>, continuity implies uniform continuity because the functional domain <i>V</i> is compact. Hence, no difference exists between the two.</p><p>The result in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e11021">16</a></sup> is described as follows:</p> <h3 class="c-article__sub-heading" id="FPar17">Theorem 3</h3> <p>(Sugiura et al.<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e11032">16</a></sup>) Let <span class="mathjax-tex">${\mathbb {F}}^*$</span> be the set of RIT operators corresponding to a uniformly continuous functional and let reservoir <span class="mathjax-tex">${\mathbb {F}}$</span> be a set of bounded RIT operators. Suppose that <span class="mathjax-tex">${\mathbb {F}}$</span> has the NSP. Then, reservoir <span class="mathjax-tex">${\mathbb {F}}$</span> is universal for uniform approximations in <span class="mathjax-tex">${\mathbb {F}}^*$</span>.</p> <p>Operator <span class="mathjax-tex">$F:U^R\rightarrow Y^R$</span> is said to be bounded if <span class="mathjax-tex">$F{\left( U^R\right) }$</span> is uniformly bounded, that is,</p><div id="Equ29" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \exists c\ge 0,\forall y\in F{\left( U\right) },\forall t\ge 0,\left| y{\left( t\right) }\right| \le c. \end{aligned}$$</span></div><div class="c-article-equation__number"> (29) </div></div><p>To summarize the result in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e11302">16</a></sup>, a reservoir with the NSP that contains only bounded operators is universal.</p><h3 class="c-article__sub-heading" id="Sec7">Existence of universal finite reservoir</h3><p>Using Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar17">3</a>, we show that a universal reservoir <span class="mathjax-tex">${\mathbb {F}}$</span> with a single output exists. If the reservoir <span class="mathjax-tex">${\mathbb {F}}$</span> is universal, we can easily construct a universal reservoir with a higher dimensional output by adding arbitrary operators to <span class="mathjax-tex">${\mathbb {F}}$</span>. Hence, the universality of <span class="mathjax-tex">${\mathbb {F}}$</span> means that, for any dimension, a universal reservoir exists with the output of that specified dimension. Let operator <span class="mathjax-tex">$F:U^R\rightarrow Y^R$</span> and functional <span class="mathjax-tex">$f:V^{\textrm{res}}\rightarrow {\mathbb {R}}$</span> correspond to each other. Then, the NSP of reservoir <span class="mathjax-tex">${\mathbb {F}}=\left\{ F\right\}$</span> is written as follows:</p><div id="Equ30" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \forall v_1,v_2\in \overline{V^{\textrm{res}}},\left[ v_1\ne v_2\Rightarrow \left( \exists \delta >0,\overline{f{\left( N_\delta {\left( v_1 \right) }\cap V^{\textrm{res}} \right) }}\cap \overline{f{\left( N_\delta {\left( v_2 \right) }\cap V^{\textrm{res}}\right) }}=\emptyset \right) \right] . \end{aligned}$$</span></div><div class="c-article-equation__number"> (30) </div></div><p>The following proposition provides a stronger but simpler condition than Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ30">30</a>):</p> <h3 class="c-article__sub-heading" id="FPar18">Proposition 3</h3> <p>Let <span class="mathjax-tex">$f:V^{\textrm{res}}\rightarrow {\mathbb {R}}$</span>. Suppose that there is a continuous left inverse <span class="mathjax-tex">$g:\overline{f{\left( V^{\textrm{res}}\right) }}\rightarrow \overline{V^{\textrm{res}}}$</span> of <i>f</i>, that is, a continuous map <i>g</i> that satisfies the following:</p><div id="Equ31" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} g\circ f=\textrm{id}_{V^{\textrm{res}}}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (31) </div></div><p>Then, <i>f</i> satisfies Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ30">30</a>).</p> <h3 class="c-article__sub-heading" id="FPar19">Proof of Proposition 3</h3> <p>We prove the contraposition. Suppose that there are distinct <span class="mathjax-tex">$v_1$</span>, <span class="mathjax-tex">$v_2\in \overline{V^{\textrm{res}}}$</span> that satisfy the following:</p><div id="Equ32" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \forall \delta >0,\overline{f{\left( V_1 \right) }}\cap \overline{f{\left( V_2\right) }}\ne \emptyset , \end{aligned}$$</span></div><div class="c-article-equation__number"> (32) </div></div><p>where <span class="mathjax-tex">$V_1$</span>, <span class="mathjax-tex">$V_2\subset V^{\textrm{res}}$</span> are defined as follows:</p><div id="Equ33" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} V_1=N_\delta {\left( v_1 \right) }\cap V^{\textrm{res}},\quad V_2=N_\delta {\left( v_2 \right) }\cap V^{\textrm{res}}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (33) </div></div><p>Because of <span class="mathjax-tex">$v_1\ne v_2$</span>, we obtain <span class="mathjax-tex">$\overline{V_1}\cap \overline{V_2}=\emptyset$</span> by setting <span class="mathjax-tex">$\delta =d{\left( v_1,v_2\right) }/3$</span>. Let <span class="mathjax-tex">$\left( \alpha _{1,i}\right) _{i\in {\mathbb {N}}}\in f{\left( V_1\right) }$</span> and <span class="mathjax-tex">$\left( \alpha _{2,i}\right) _{i\in {\mathbb {N}}}\in f{\left( V_2\right) }$</span> be sequences that converge to <span class="mathjax-tex">$\alpha \in \overline{f{\left( V_1 \right) }}\cap \overline{f{\left( V_2\right) }}$</span>. Suppose that <span class="mathjax-tex">$g:\overline{f{\left( V^{\textrm{res}}\right) }}\rightarrow \overline{V^{\textrm{res}}}$</span> is a left inverse of <i>f</i>. Then, <span class="mathjax-tex">$g{\left( \alpha _{1,i}\right) }$</span> and <span class="mathjax-tex">$g{\left( \alpha _{2,i}\right) }$</span> are contained in <span class="mathjax-tex">$V_1$</span> and <span class="mathjax-tex">$V_2$</span>, respectively. Because of <span class="mathjax-tex">$\overline{V_1}\cap \overline{V_2}=\emptyset$</span>, <span class="mathjax-tex">$g{\left( \alpha _{1,i}\right) }$</span> and <span class="mathjax-tex">$g{\left( \alpha _{2,i}\right) }$</span> do not converge to the same point as <span class="mathjax-tex">$i\rightarrow \infty$</span>, even though <span class="mathjax-tex">$\alpha _{1,i}$</span> and <span class="mathjax-tex">$\alpha _{2,i}$</span> converge to <span class="mathjax-tex">$\alpha$</span>. Hence, <i>g</i> is not continuous at <span class="mathjax-tex">$\alpha \in \overline{f{\left( V^{\textrm{res}}\right) }}$</span>. <span class="mathjax-tex">$\square$</span></p> <p>To prove that a functional <span class="mathjax-tex">$f:V^{\textrm{res}}\rightarrow {\mathbb {R}}$</span> with a continuous left inverse exists, we use the following theorem.</p> <h3 class="c-article__sub-heading" id="FPar20">Theorem 4</h3> <p>(Hahn–Mazurkiewicz theorem<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Hocking, J. G. & Young, G. S. Topology (Addison-Wesley Publishing Company, 1961)." href="/article/10.1038/s41598-024-56742-7#ref-CR20" id="ref-link-section-d117352412e13013">20</a></sup>) Let <i>E</i> be a connected, locally connected, and compact metric space. Then, a continuous surjection from <span class="mathjax-tex">$\left[ 0,1\right]$</span> to <i>E</i> exists.</p> <p>Metric space <span class="mathjax-tex">$\overline{V^{\textrm{res}}}=V^{\textrm{res}}\cup V$</span> is compact<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e13094">16</a></sup>. As we show later in this paper, <span class="mathjax-tex">$\overline{V^{\textrm{res}}}$</span> is connected and locally connected. Hence, from Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar20">4</a>, a space-filling curve <span class="mathjax-tex">$g:\left[ 0,1\right] \rightarrow \overline{V^{\textrm{res}}}$</span> exists, which is continuous and surjective. Using the axiom of choice, we obtain a functional <span class="mathjax-tex">$f:V^{\textrm{res}}\rightarrow {\mathbb {R}}$</span> with a continuous left inverse as a right inverse of <i>g</i>. Therefore, a reservoir that contains only one operator can have the NSP. To prove that <span class="mathjax-tex">$\overline{V^{\textrm{res}}}$</span> is connected and locally connected, we assume that the set <span class="mathjax-tex">$A\subset {\mathbb {R}}^n$</span> in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ8">8</a>) is convex.</p> <h3 class="c-article__sub-heading" id="FPar21">Proposition 4</h3> <p>The set <span class="mathjax-tex">$\overline{V^{\textrm{res}}}$</span> is connected.</p> <h3 class="c-article__sub-heading" id="FPar22">Proof of Proposition 4</h3> <p>We prove that <span class="mathjax-tex">$\overline{V^{{\textrm{res}}}}$</span> is arcwise connected, which is a stronger condition than being connected. Let <span class="mathjax-tex">$v_1$</span> and <span class="mathjax-tex">$v_2$</span> be arbitrary inputs in <span class="mathjax-tex">$\overline{V^{{\textrm{res}}}}$</span>. The set <span class="mathjax-tex">$\overline{V^{\textrm{res}}}$</span> is said to be arcwise connected if there is some continuous map <span class="mathjax-tex">$P:\left[ 0,1\right] \rightarrow \overline{V^{{\textrm{res}}}}$</span> that satisfies <span class="mathjax-tex">$P{\left( 0\right) }=v_1$</span> and <span class="mathjax-tex">$P{\left( 1\right) }=v_2$</span>. Such a map <i>P</i> is called a path from <span class="mathjax-tex">$v_1$</span> to <span class="mathjax-tex">$v_2$</span>. Let <span class="mathjax-tex">$t_1=\lambda {\left( v_1\right) }$</span> and <span class="mathjax-tex">$t_2=\lambda {\left( v_2\right) }$</span>. If a path exists from <span class="mathjax-tex">$v_1$</span> to <span class="mathjax-tex">$v_2$</span>, one exists from <span class="mathjax-tex">$v_2$</span> to <span class="mathjax-tex">$v_1$</span>. Hence, we can assume that <span class="mathjax-tex">$t_1\le t_2$</span> without loss of generality. Let <span class="mathjax-tex">$v_3=v_2^{\left[ t_1\right] }$</span>, that is, <span class="mathjax-tex">$v_3$</span> has the same domain as <span class="mathjax-tex">$v_1$</span> and the same value as <span class="mathjax-tex">$v_2$</span>. We form a path from <span class="mathjax-tex">$v_1$</span> to <span class="mathjax-tex">$v_2$</span> through <span class="mathjax-tex">$v_3$</span>.</p> <p>First, we form a path from <span class="mathjax-tex">$v_1$</span> to <span class="mathjax-tex">$v_3$</span>. Because <span class="mathjax-tex">$v_1$</span> and <span class="mathjax-tex">$v_3$</span> has the same domain <span class="mathjax-tex">$\left[ -t_1,0\right]$</span>, we can define a map <span class="mathjax-tex">$P_1:\left[ 0,1\right] \rightarrow \overline{V^{\textrm{res}}}$</span> as follows:</p><div id="Equ34" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} P_1{\left( \alpha \right) }=\left( 1-\alpha \right) v_1+\alpha v_3\quad \left( \alpha \in \left[ 0,1\right] \right) . \end{aligned}$$</span></div><div class="c-article-equation__number"> (34) </div></div><p>Map <span class="mathjax-tex">$P_1$</span> satisfies <span class="mathjax-tex">$P_1{\left( 0\right) }=v_1$</span> and <span class="mathjax-tex">$P_1{\left( 1\right) }=v_3$</span>. Because the input range <span class="mathjax-tex">$A\subset {\mathbb {R}}^n$</span> is convex, the image of <span class="mathjax-tex">$P_1$</span> is included in <span class="mathjax-tex">$\overline{V^{\textrm{res}}}$</span>. Map <span class="mathjax-tex">$P_1$</span> is continuous because, for any <span class="mathjax-tex">$\alpha _1$</span>, <span class="mathjax-tex">$\alpha _2\in \left[ 0,1\right]$</span>, the distance between <span class="mathjax-tex">$P_1{\left( \alpha _1\right) }$</span> and <span class="mathjax-tex">$P_1{\left( \alpha _2\right) }$</span> is formulated as follows:</p><div id="Equ35" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} d{\left( P_1{\left( \alpha _1\right) },P_1{\left( \alpha _2\right) }\right) }=\left| \alpha _1-\alpha _2\right| \left\| v_1-v_3\right\| _w. \end{aligned}$$</span></div><div class="c-article-equation__number"> (35) </div></div><p>Therefore, map <span class="mathjax-tex">$P_1$</span> is a path from <span class="mathjax-tex">$v_1$</span> to <span class="mathjax-tex">$v_3$</span>.</p> <p>Second, we form a path from <span class="mathjax-tex">$v_3$</span> to <span class="mathjax-tex">$v_2$</span>. The function <span class="mathjax-tex">$\theta$</span> has inverse <span class="mathjax-tex">$\theta ^{-1}$</span> because it is strictly increasing and continuous. Using <span class="mathjax-tex">$\theta ^{-1}$</span>, we define map <span class="mathjax-tex">$\mu :\left[ 0,1\right] \rightarrow \left[ t_1,t_2\right]$</span> as follows:</p><div id="Equ36" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \mu {\left( \alpha \right) }=\theta ^{-1}{\left( \left( 1-\alpha \right) \theta {\left( t_1\right) }+\alpha \theta {\left( t_2\right) }\right) }\quad \left( \alpha \in \left[ 0,1\right] \right) , \end{aligned}$$</span></div><div class="c-article-equation__number"> (36) </div></div><p>where <span class="mathjax-tex">$\theta {\left( \infty \right) }=\lim _{t\rightarrow \infty }\theta {\left( t\right) }$</span> and <span class="mathjax-tex">$\theta ^{-1}{\left( \theta {\left( \infty \right) }\right) }=\infty$</span>. We define a map <span class="mathjax-tex">$P_2:\left[ 0,1\right] \rightarrow \overline{V^{\textrm{res}}}$</span> as follows:</p><div id="Equ37" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} P_2{\left( \alpha \right) }=v_2^{\left[ \mu {\left( \alpha \right) }\right] }\quad \left( \alpha \in \left[ 0,1\right] \right) . \end{aligned}$$</span></div><div class="c-article-equation__number"> (37) </div></div><p>Map <span class="mathjax-tex">$P_2$</span> satisfies <span class="mathjax-tex">$P_2{\left( 0\right) }=v_3$</span> and <span class="mathjax-tex">$P_2{\left( 1\right) }=v_2$</span> because <span class="mathjax-tex">$\mu {\left( 0\right) }=t_1$</span> and <span class="mathjax-tex">$\mu {\left( 1\right) }=t_2$</span> hold. From the definition (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ21">21</a>) of <span class="mathjax-tex">$V^{\textrm{res}}$</span>, the image of <span class="mathjax-tex">$P_2$</span> is included in <span class="mathjax-tex">$\overline{V^{\textrm{res}}}$</span>. Map <span class="mathjax-tex">$P_2$</span> is continuous because, for any <span class="mathjax-tex">$\alpha _1$</span>, <span class="mathjax-tex">$\alpha _2\in \left[ 0,1\right]$</span>, the distance between <span class="mathjax-tex">$P_2{\left( \alpha _1\right) }$</span> and <span class="mathjax-tex">$P_2{\left( \alpha _2\right) }$</span> is formulated as follows:</p><div id="Equ38" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} d{\left( P_2{\left( \alpha _1\right) },P_2{\left( \alpha _2\right) }\right) }=\left| \theta {\left( \mu {\left( \alpha _1\right) }\right) }-\theta {\left( \mu {\left( \alpha _2\right) }\right) }\right| =\left| \alpha _1-\alpha _2\right| \left| \theta {\left( t_1\right) }-\theta {\left( t_2\right) }\right| . \end{aligned}$$</span></div><div class="c-article-equation__number"> (38) </div></div><p>Therefore, map <span class="mathjax-tex">$P_2$</span> is a path from <span class="mathjax-tex">$v_3$</span> to <span class="mathjax-tex">$v_2$</span>.</p> <p>Using <span class="mathjax-tex">$P_1$</span> and <span class="mathjax-tex">$P_2$</span>, we define path <span class="mathjax-tex">$P:\left[ 0,1\right] \rightarrow \overline{V^{\textrm{res}}}$</span> from <span class="mathjax-tex">$v_1$</span> to <span class="mathjax-tex">$v_2$</span> as follows:</p><div id="Equ39" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} P{\left( \alpha \right) }= {\left\{ \begin{array}{ll} P_1{\left( 2\alpha \right) }&{}\left( 0\le \alpha \le \frac{1}{2}\right) \\ P_2{\left( 2\alpha -1\right) }&{}\left( \frac{1}{2}<\alpha \le 1\right) \end{array}\right. }. \end{aligned}$$</span></div><div class="c-article-equation__number"> (39) </div></div><p>Therefore, the set <span class="mathjax-tex">$\overline{V^{\textrm{res}}}$</span> is arcwise connected. <span class="mathjax-tex">$\square$</span></p> <p>To prove that the set <span class="mathjax-tex">$\overline{V^{\textrm{res}}}$</span> is locally connected, we use the following proposition:</p> <h3 class="c-article__sub-heading" id="FPar23">Proposition 5</h3> <p>For any <span class="mathjax-tex">$v_1$</span>, <span class="mathjax-tex">$v_2\in \overline{V^{\textrm{res}}}$</span>, path <span class="mathjax-tex">$P:\left[ 0,1\right] \rightarrow \overline{V^{\textrm{res}}}$</span> from <span class="mathjax-tex">$v_1$</span> to <span class="mathjax-tex">$v_2$</span> exists that satisfies the following:</p><div id="Equ40" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \forall \alpha \in \left[ 0,1\right] ,d{\left( v_1,v_2\right) }=d{\left( v_1,P{\left( \alpha \right) }\right) }+d{\left( P{\left( \alpha \right) },v_2\right) }. \end{aligned}$$</span></div><div class="c-article-equation__number"> (40) </div></div> <p>Equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ40">40</a>) means that path <i>P</i> is “the shortest” between <span class="mathjax-tex">$v_1$</span> and <span class="mathjax-tex">$v_2$</span>, that is, equality holds in the triangle inequality between any point on the path and the two endpoints.</p> <h3 class="c-article__sub-heading" id="FPar24">Proof of Proposition 5</h3> <p>Let <span class="mathjax-tex">$v_1$</span> and <span class="mathjax-tex">$v_2$</span> be arbitrary inputs in <span class="mathjax-tex">$\overline{V^{\textrm{res}}}$</span>. If path <i>P</i> from <span class="mathjax-tex">$v_1$</span> to <span class="mathjax-tex">$v_2$</span> exists that satisfies Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ40">40</a>), a similar path from <span class="mathjax-tex">$v_2$</span> to <span class="mathjax-tex">$v_1$</span> exists. Hence, for <span class="mathjax-tex">$t_1=\lambda {\left( v_1\right) }$</span> and <span class="mathjax-tex">$t_2=\lambda {\left( v_2\right) }$</span>, we can assume that <span class="mathjax-tex">$t_1\le t_2$</span> without loss of generality. We show that path <span class="mathjax-tex">$P:\left[ 0,1\right] \rightarrow \overline{V^{\textrm{res}}}$</span> defined by (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ39">39</a>) satisfies Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ40">40</a>) using the following three lemmas: <span class="mathjax-tex">$\square$</span></p> <h3 class="c-article__sub-heading" id="FPar25">Lemma 2</h3> <p>Path <span class="mathjax-tex">$P_1:\left[ 0,1\right] \rightarrow \overline{V^{\textrm{res}}}$</span> defined by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ34">34</a>) satisfies the following:</p><div id="Equ41" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \forall \alpha \in \left[ 0,1\right] ,d{\left( v_1,v_3\right) }=d{\left( v_1,P_1{\left( \alpha \right) }\right) }+d{\left( P_1{\left( \alpha \right) },v_3\right) }. \end{aligned}$$</span></div><div class="c-article-equation__number"> (41) </div></div> <h3 class="c-article__sub-heading" id="FPar26">Proof of Lemma 2</h3> <p>Using Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ35">35</a>), distance <span class="mathjax-tex">$d{\left( v_1,v_3\right) }$</span> is transformed as follows:</p><div id="Equ42" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{aligned} d{\left( v_1,v_3\right) }=&\,\left\| v_1-v_3\right\| _w\\ =&\,\left| 0-\alpha \right| \left\| v_1-v_3\right\| _w+\left| \alpha -1\right| \left\| v_1-v_3\right\| _w\\ =&\,d{\left( P_1{\left( 0\right) },P_1{\left( \alpha \right) }\right) }+d{\left( P_1{\left( \alpha \right) },P_1{\left( 1\right) }\right) }\\ =&\,d{\left( v_1,P_1{\left( \alpha \right) }\right) }+d{\left( P_1{\left( \alpha \right) },v_3\right) }, \end{aligned} \end{aligned}$$</span></div><div class="c-article-equation__number"> (42) </div></div><p>which proves Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar25">2</a>. <span class="mathjax-tex">$\square$</span></p> <h3 class="c-article__sub-heading" id="FPar27">Lemma 3</h3> <p>Path <span class="mathjax-tex">$P_2:\left[ 0,1\right] \rightarrow \overline{V^{\textrm{res}}}$</span> defined by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ37">37</a>) satisfies the following:</p><div id="Equ43" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \forall \alpha \in \left[ 0,1\right] ,d{\left( v_3,v_2\right) }=d{\left( v_3,P_2{\left( \alpha \right) }\right) }+d{\left( P_2{\left( \alpha \right) },v_2\right) }. \end{aligned}$$</span></div><div class="c-article-equation__number"> (43) </div></div> <h3 class="c-article__sub-heading" id="FPar28">Proof of Lemma 3</h3> <p>Using Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ38">38</a>), distance <span class="mathjax-tex">$d{\left( v_3,v_2\right) }$</span> is transformed as follows:</p><div id="Equ44" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{aligned} d{\left( v_3,v_2\right) }=&\,\left| \theta {\left( t_1\right) }-\theta {\left( t_2\right) }\right| \\ =&\,\left| 0-\alpha \right| \left| \theta {\left( t_1\right) }-\theta {\left( t_2\right) }\right| +\left| \alpha -1\right| \left| \theta {\left( t_1\right) }-\theta {\left( t_2\right) }\right| \\ =&\,d{\left( P_2{\left( 0\right) },P_2{\left( \alpha \right) }\right) }+d{\left( P_2{\left( \alpha \right) },P_2{\left( 1\right) }\right) }\\ =&\,d{\left( v_3,P_2{\left( \alpha \right) }\right) }+d{\left( P_2{\left( \alpha \right) },v_2\right) }, \end{aligned} \end{aligned}$$</span></div><div class="c-article-equation__number"> (44) </div></div><p>which proves Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar27">3</a>. <span class="mathjax-tex">$\square$</span></p> <h3 class="c-article__sub-heading" id="FPar29">Lemma 4</h3> <p>The input <span class="mathjax-tex">$v_3=v_2^{\left[ t_1\right] }$</span> satisfies the following:</p><div id="Equ45" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} d{\left( v_1,v_2\right) }=d{\left( v_1,v_3\right) }+d{\left( v_3,v_2\right) }. \end{aligned}$$</span></div><div class="c-article-equation__number"> (45) </div></div> <h3 class="c-article__sub-heading" id="FPar30">Proof of Lemma 4</h3> <p>Distance <span class="mathjax-tex">$d{\left( v_1,v_2\right) }$</span> is transformed as follows:</p><div id="Equ46" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{aligned} d{\left( v_1,v_2\right) }=&\,\left\| v_1-v_2^{\left[ t_1\right] }\right\| _w+\left| \theta {\left( t_1\right) }-\theta {\left( t_2\right) }\right| \\ =&\,\left\| v_1-v_3\right\| _w+\left| \theta {\left( \lambda {\left( v_3\right) }\right) }-\theta {\left( t_2\right) }\right| \\ =&\,d{\left( v_1,v_3\right) }+d{\left( v_3,v_2\right) }, \end{aligned} \end{aligned}$$</span></div><div class="c-article-equation__number"> (46) </div></div><p>which proves Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar29">4</a>. <span class="mathjax-tex">$\square$</span></p> <p>Let <span class="mathjax-tex">$\alpha$</span> be an arbitrary number in <span class="mathjax-tex">$\left[ 0,1\right]$</span>. We show that the following holds:</p><div id="Equ47" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} d{\left( v_1,P{\left( \alpha \right) }\right) }+d{\left( P{\left( \alpha \right) },v_2\right) }\le d{\left( v_1,v_2\right) }. \end{aligned}$$</span></div><div class="c-article-equation__number"> (47) </div></div><p>If <span class="mathjax-tex">$0\le \alpha \le \frac{1}{2}$</span>, we obtain Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ47">47</a>) as follows:</p><div id="Equ48" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} d{\left( v_1,P{\left( \alpha \right) }\right) }+d{\left( P{\left( \alpha \right) },v_2\right) }\le d{\left( v_1,P{\left( \alpha \right) }\right) }+d{\left( P{\left( \alpha \right) },v_3\right) }+d{\left( v_3,v_2\right) } = d{\left( v_1,v_3\right) }+d{\left( v_3,v_2\right) } =d{\left( v_1,v_2\right) }. \end{aligned}$$</span></div><div class="c-article-equation__number"> (48) </div></div><p>The inequality is a triangle inequality. The first equality is obtained from Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar25">2</a> because <span class="mathjax-tex">$P{\left( \alpha \right) }=P_1{\left( 2\alpha \right) }$</span> holds. The second equality is obtained from Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar29">4</a>. If <span class="mathjax-tex">$\frac{1}{2}<\alpha \le 1$</span>, we obtain Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ47">47</a>) as follows:</p><div id="Equ49" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} d{\left( v_1,P{\left( \alpha \right) }\right) }+d{\left( P{\left( \alpha \right) },v_2\right) }\le d{\left( v_1,v_3\right) }+d{\left( v_3,P{\left( \alpha \right) }\right) }+d{\left( P{\left( \alpha \right) },v_2\right) } =d{\left( v_1,v_3\right) }+d{\left( v_3,v_2\right) } =d{\left( v_1,v_2\right) }. \end{aligned}$$</span></div><div class="c-article-equation__number"> (49) </div></div><p>The inequality is a triangle inequality. The first equality is obtained from Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar27">3</a> because <span class="mathjax-tex">$P{\left( \alpha \right) }=P_2{\left( 2\alpha -1\right) }$</span> holds. The second equality is obtained from Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar29">4</a>. From Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ47">47</a>) and the triangle inequality <span class="mathjax-tex">$d{\left( v_1,v_2\right) }\le d{\left( v_1,P{\left( \alpha \right) }\right) }+d{\left( P{\left( \alpha \right) },v_2\right) }$</span>, we obtain Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ40">40</a>), which proves Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar23">5</a>. <span class="mathjax-tex">$\square$</span></p><p>Using Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar23">5</a>, we prove the following proposition:</p> <h3 class="c-article__sub-heading" id="FPar31">Proposition 6</h3> <p>The set <span class="mathjax-tex">$\overline{V^{\textrm{res}}}$</span> is locally connected.</p> <h3 class="c-article__sub-heading" id="FPar32">Proof of Proposition 6</h3> <p>We prove that <span class="mathjax-tex">$\overline{V^{\textrm{res}}}$</span> is locally arcwise connected, which is a stronger condition than being locally connected. Let <i>v</i> be an arbitrary input in <span class="mathjax-tex">$\overline{V^{\textrm{res}}}$</span>. The set <span class="mathjax-tex">$\overline{V^{\textrm{res}}}$</span> is said to be locally arcwise connected if, for any open set <span class="mathjax-tex">$V_1\subset \overline{V^{\textrm{res}}}$</span> including <i>v</i>, an arcwise connected open set <span class="mathjax-tex">$V_2\subset V_1$</span> exists that includes <i>v</i>. Therefore, it is sufficient for the Proof of Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar31">6</a> to show that a <span class="mathjax-tex">$\delta$</span>-neighborhood <span class="mathjax-tex">$N_\delta {\left( v\right) }\subset \overline{V^{\textrm{res}}}$</span> of <i>v</i> is arcwise connected for any <span class="mathjax-tex">$\delta >0$</span>. Let <span class="mathjax-tex">$\delta$</span> be an arbitrary positive number and <span class="mathjax-tex">$v_1$</span>, <span class="mathjax-tex">$v_2$</span> be arbitrary inputs in <span class="mathjax-tex">$N_\delta {\left( v\right) }$</span>. From Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar23">5</a> and <span class="mathjax-tex">$d{\left( v,v_1\right) }<\delta$</span>, path <span class="mathjax-tex">$P_1:\left[ 0,1\right] \rightarrow \overline{V^{\textrm{res}}}$</span> from <span class="mathjax-tex">$v_1$</span> to <i>v</i> exists that satisfies the following for any <span class="mathjax-tex">$\alpha \in \left[ 0,1\right]$</span>:</p><div id="Equ50" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} d{\left( v,P_1{\left( \alpha \right) }\right) }=d{\left( v,v_1\right) }-d{\left( P_1{\left( \alpha \right) },v_1\right) }<\delta -d{\left( P_1{\left( \alpha \right) },v_1\right) }\le \delta . \end{aligned}$$</span></div><div class="c-article-equation__number"> (50) </div></div><p>Hence, <span class="mathjax-tex">$P_1{\left( \left[ 0,1\right] \right) }\subset N_\delta {\left( v\right) }$</span> holds. Similarly, we obtain path <span class="mathjax-tex">$P_2:\left[ 0,1\right] \rightarrow N_\delta {\left( v\right) }$</span> from <i>v</i> to <span class="mathjax-tex">$v_2$</span>. Therefore, path <span class="mathjax-tex">$P:\left[ 0,1\right] \rightarrow N_\delta {\left( v\right) }$</span> from <span class="mathjax-tex">$v_1$</span> to <span class="mathjax-tex">$v_2$</span> exists, which proves Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar31">6</a>. <span class="mathjax-tex">$\square$</span></p> <p>Using Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar17">3</a> and Propositions <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar18">3</a>, <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar21">4</a>, and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar31">6</a>, we show that a universal reservoir exists that has only one output.</p> <h3 class="c-article__sub-heading" id="FPar33">Theorem 5</h3> <p>Let <span class="mathjax-tex">${\mathbb {F}}^*$</span> be the set of RIT operators corresponding to a uniformly continuous functional. Assume the axiom of choice and that the set <span class="mathjax-tex">$A\subset {\mathbb {R}}^n$</span> in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ8">8</a>) is convex. Then, reservoir <span class="mathjax-tex">${\mathbb {F}}$</span> exists that is a set of only one RIT operator and <span class="mathjax-tex">${\mathbb {F}}$</span> is universal for uniform approximations in <span class="mathjax-tex">${\mathbb {F}}^*$</span>.</p> <h3 class="c-article__sub-heading" id="FPar34">Proof of Theorem 5</h3> <p>Metric space <span class="mathjax-tex">$\overline{V^{\textrm{res}}}=V^{\textrm{res}}\cup V$</span> is compact<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e20724">16</a></sup>. From Propositions <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar21">4</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar31">6</a>, <span class="mathjax-tex">$\overline{V^{\textrm{res}}}$</span> is connected and locally connected. Hence, from Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar20">4</a>, a continuous surjection <span class="mathjax-tex">$g:\left[ 0,1\right] \rightarrow \overline{V^{\textrm{res}}}$</span> exists. We define a set <span class="mathjax-tex">$S\subset \left[ 0,1\right]$</span> as follows:</p><div id="Equ51" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} S=\left\{ \alpha \in \left[ 0,1\right] \left| g{\left( \alpha \right) }\in V^{\textrm{res}}\right. \right\} . \end{aligned}$$</span></div><div class="c-article-equation__number"> (51) </div></div><p>Let <span class="mathjax-tex">$g|_S:S\rightarrow V^{\textrm{res}}$</span> be a restriction of <i>g</i> to <i>S</i>. Because <span class="mathjax-tex">$g|_S$</span> is a surjection to <span class="mathjax-tex">$V^{\textrm{res}}$</span>, the axiom of choice provides a right inverse <span class="mathjax-tex">$f:V^{\textrm{res}}\rightarrow S$</span> of <span class="mathjax-tex">$g|_S$</span>, that is, <i>f</i> satisfies <span class="mathjax-tex">$g|_S\circ f=\textrm{id}_{V^{\textrm{res}}}$</span>. Functional <i>f</i> and an another restriction <span class="mathjax-tex">$g|_{\overline{S}}:\overline{S}\rightarrow \overline{V^{\textrm{res}}}$</span> of <i>g</i> also satisfy <span class="mathjax-tex">$g|_{\overline{S}}\circ f=\textrm{id}_{V^{\textrm{res}}}$</span>. Because the restriction <span class="mathjax-tex">$g|_{\overline{S}}$</span> is continuous, from Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar18">3</a>, <i>f</i> satisfies Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1038/s41598-024-56742-7#Equ30">30</a>). Hence, reservoir <span class="mathjax-tex">${\mathbb {F}}=\left\{ F\right\}$</span> has the NSP, where <span class="mathjax-tex">$F:U^R\rightarrow Y^R$</span> is the corresponding operator of <i>f</i>. Because of <span class="mathjax-tex">$S\subset \left[ 0,1\right]$</span>, functional <i>f</i> and operator <i>F</i> are bounded. Therefore, reservoir <span class="mathjax-tex">${\mathbb {F}}$</span> satisfies the condition of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar17">3</a> and is universal for uniform approximations in <span class="mathjax-tex">${\mathbb {F}}^*$</span>. <span class="mathjax-tex">$\square$</span></p> <p>Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar33">5</a> shows that a universal finite reservoir exists, which achieves the low computational cost of training. Moreover, a universal reservoir exists independent of the dimension of its output. The reservoir obtained from a right inverse of a space-filling curve proposed in the Proof of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar33">5</a> is probably chaotic. This result suggests that chaos is key to the reservoir’s universality. In practice, chaotic reservoirs have already been considered in research and are known to be useful<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Jensen, J. H. & Tufte, G. Reservoir computing with a chaotic circuit. In Artificial Life Conference Proceedings 222–229 (MIT Press, 2017)." href="/article/10.1038/s41598-024-56742-7#ref-CR21" id="ref-link-section-d117352412e21478">21</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="Choi, J. & Kim, P. Reservoir computing based on quenched chaos. Chaos Solitons Fractals 140, 110131 (2020)." href="/article/10.1038/s41598-024-56742-7#ref-CR22" id="ref-link-section-d117352412e21481">22</a></sup>.</p><p>The difference between the results of Sections “<a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1038/s41598-024-56742-7#Sec2">Reservoir computing represented by bi-infinite-time operators</a>” and “<a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1038/s41598-024-56742-7#Sec5">Reservoir computing represented by right-infinite-time operators</a>” is not caused by the difference between the BIT and RIT operators but the difference between the conditions of functionals <span class="mathjax-tex">$f_1,\ldots ,f_m$</span> corresponding to operators in the reservoir. Fading memory and the separation property, which are the conditions of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar4">1</a>, require the map <span class="mathjax-tex">$\left( f_1,\ldots ,f_m\right)$</span> to be continuous and injective; however, this is impossible. By contrast, the NSP, which is the condition of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1038/s41598-024-56742-7#FPar17">3</a>, only requires the “strong injectivity” of <span class="mathjax-tex">$\left( f_1,\ldots ,f_m\right)$</span>, which is possible as described thus far. The NSP is also defined for a reservoir represented by BIT operators as follows:</p> <h3 class="c-article__sub-heading" id="FPar35">Definition 10</h3> <p>Let <span class="mathjax-tex">${\mathbb {F}}$</span> be a set of causal BIT operators <span class="mathjax-tex">$F_1,\ldots ,F_m$</span> and let <span class="mathjax-tex">$f_1,\ldots ,f_m:V\rightarrow {\mathbb {R}}$</span> correspond to each operator in <span class="mathjax-tex">${\mathbb {F}}$</span>. The set <span class="mathjax-tex">${\mathbb {F}}$</span> is said to have the NSP if the following holds for any distinct <span class="mathjax-tex">$v_1$</span>, <span class="mathjax-tex">$v_2\in V$</span>:</p><div id="Equ52" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \exists \delta >0,\overline{\left( f_1,\ldots ,f_m\right) {\left( N_\delta {\left( v_1 \right) }\right) }}\cap \overline{\left( f_1,\ldots ,f_m\right) {\left( N_\delta {\left( v_2 \right) }\right) }}=\emptyset , \end{aligned}$$</span></div><div class="c-article-equation__number"> (52) </div></div><p>where <span class="mathjax-tex">$N_\delta {\left( v \right) }\subset V$</span> is a <span class="mathjax-tex">$\delta$</span>-neighborhood of <span class="mathjax-tex">$v\in V$</span>.</p> <p>A finite reservoir with the NSP of Definition 10 can be obtained in the same manner as in the RIT operator case. Using Theorem 4 in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e22049">16</a></sup>, that is, an extension of the Stone–Weierstrass theorem, we can derive the universality of Definition 1 from the NSP of Definition 10. Therefore, the NSP is the essence of proving the existence of a universal finite reservoir.</p></div></div></section><section data-title="Conclusion"><div class="c-article-section" id="Sec8-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec8">Conclusion</h2><div class="c-article-section__content" id="Sec8-content"><p>We discussed whether a universal finite reservoir exists under the assumption that the readout is a polynomial and the target operator has fading memory. In the discussion, we considered two sufficient conditions for the reservoir to be universal proposed by Maass et al.<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e22062">2</a></sup> and in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e22066">16</a></sup>. First, we showed that no finite reservoir satisfies the condition in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Maass, W. & Natschl, T. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput. 14(11), 2531–2560 (2002)." href="/article/10.1038/s41598-024-56742-7#ref-CR2" id="ref-link-section-d117352412e22070">2</a></sup>. Supposing that such a reservoir exists, we derived the contradiction that the input function space and a subset of finite-dimensional vector space are homeomorphic. Next, we proposed an example of a universal reservoir that has a single output and the NSP, which is the condition in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S. Nonessentiality of reservoir’s fading memory for universality of reservoir computing. IEEE Trans. Neural Netw. 
 https://doi.org/10.1109/TNNLS.2023.3298013
 
 (2023)." href="/article/10.1038/s41598-024-56742-7#ref-CR16" id="ref-link-section-d117352412e22074">16</a></sup>. The functional corresponding to the operator representing the reservoir has a continuous left inverse. We showed that this is a stronger condition than the NSP. The functional is defined as a right inverse of the continuous surjection from <span class="mathjax-tex">$\left[ 0,1\right]$</span> to the input space of the functional. The surjection is given by the Hahn–Mazurkiewicz theorem. Our example means that, for any dimension, a universal reservoir exists that has the output of that specified dimension. 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id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">Sugiura, S., Ariizumi, R., Asai, T. <i>et al.</i> Existence of reservoir with finite-dimensional output for universal reservoir computing. <i>Sci Rep</i> <b>14</b>, 8448 (2024). https://doi.org/10.1038/s41598-024-56742-7</p><p class="c-bibliographic-information__download-citation u-hide-print"><a data-test="citation-link" data-track="click" data-track-action="download article citation" data-track-label="link" data-track-external="" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1038/s41598-024-56742-7?format=refman&flavour=citation">Download citation<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p><ul class="c-bibliographic-information__list" data-test="publication-history"><li 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