Document Zbl 07843840

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<a href="/authors/xu.bo.3" title="Author Profile">Xu, Bo</a><a class="btn btn-default btn-xs matrix" type="button" href="https://donald.zentralblatt-math.org/lethe/recruit/?first_name=Bo&last_name=Xu&document_id=7843840"> <span class="text-danger">Recruit</span> </a>; <a href="/authors/cao.shuhao" title="Author Profile">Cao, Shuhao</a><a class="btn btn-default btn-xs matrix" type="button" href="https://donald.zentralblatt-math.org/lethe/recruit/?first_name=Shuhao&last_name=Cao&author_id=cao.shuhao&document_id=7843840"> <span class="text-danger">Recruit</span> </a>; <a href="/authors/zhang.lei.5" title="Author Profile">Zhang, Lei</a><a class="btn btn-default btn-xs matrix" type="button" href="https://donald.zentralblatt-math.org/lethe/recruit/?first_name=Lei&last_name=Zhang&author_id=zhang.lei.5&document_id=7843840"> <span class="text-danger">Recruit</span> </a></div> <h2 class="title"> <strong>Mitigating spectral bias for the multiscale operator learning.</strong> <i>(English)</i> <a class="label nowrap" href="/7843840">Zbl 07843840</a> </h2> <div class="source"> <a href="/serials/127" title="Journal Profile">J. Comput. Phys.</a> <a href="/?q=in%3A511902" title="Articles in this Issue">506, Article ID 112944, 18 p. (2024)</a>. </div> <div class="abstract"><div class="pre">Summary: Neural operators have emerged as a powerful tool for learning the mapping between infinite-dimensional parameter and solution spaces of partial differential equations (PDEs). In this work, we focus on multiscale PDEs that have important applications such as reservoir modeling and turbulence prediction. We demonstrate that for such PDEs, the spectral bias towards low-frequency components presents a significant challenge for existing neural operators. To address this challenge, we propose a hierarchical attention neural operator (HANO) inspired by the hierarchical matrix approach. HANO features a scale-adaptive interaction range and self-attentions over a hierarchy of levels, enabling nested feature computation with controllable linear cost and encoding/decoding of multiscale solution space. We also incorporate an empirical $H^1$ loss function to enhance the learning of high-frequency components. Our numerical experiments demonstrate that HANO outperforms state-of-the-art (SOTA) methods for representative multiscale problems.</div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7843840">Cited in <strong>2</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65Nxx" title="MSC2020">65Nxx</a> </td> <td class="space"> Numerical methods for partial differential equations, boundary value problems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68Txx" title="MSC2020">68Txx</a> </td> <td class="space"> Artificial intelligence </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65Fxx" title="MSC2020">65Fxx</a> </td> <td class="space"> Numerical linear algebra </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Apartial+differential+equations">partial differential equations</a>; <a href="/?q=ut%3Aoperator+learning">operator learning</a>; <a href="/?q=ut%3Atransformer">transformer</a>; <a href="/?q=ut%3Amultiscale+PDE">multiscale PDE</a></div> <div class="software"> <h3>Software:</h3><a href="/software/42093">DeepONet</a>; <a href="/software/42135">GPT-3</a>; <a href="/software/45343">FMMformer</a>; <a href="/software/33176">U-Net</a>; <a href="/software/45340">Nyströmformer</a>; <a href="/software/50130">GNOT</a>; <a href="/software/42544">ViT</a>; <a href="/software/26507">Tensor2Tensor</a>; <a href="/software/42564">Swin Transformer</a>; <a href="/software/44700">Linformer</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 07843840" data-ciurl="/ci/07843840" data-biburl="/bibtex/07843840.bib" data-amsurl="/amsrefs/07843840.bib" data-xmlurl="/xml/07843840.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07843840.pdf" title="Zbl 07843840 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.jcp.2024.112944" aria-label="DOI for “Mitigating spectral bias for the multiscale operator learning”" title="10.1016/j.jcp.2024.112944">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2210.10890" title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Branets, L. V.; Ghai, S. S.; Wu, X.-H., Challenges and technologies in reservoir modeling, Commun. Comput. Phys., 6, 1, 1-23, 2009 · <a href="/1364.76216" class="nowrap">Zbl 1364.76216</a></td> </tr><tr> <td>[2]</td> <td class="space">Engquist, B.; Souganidis, P. E., Asymptotic and numerical homogenization, Acta Numer., 17, 147-190, 2008 · <a href="/1179.65142" class="nowrap">Zbl 1179.65142</a></td> </tr><tr> <td>[3]</td> <td class="space">Hou, T. Y.; Wu, X.-H.; Cai, Z., Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comput., 68, 227, 913-943, 1999 · <a href="/0922.65071" class="nowrap">Zbl 0922.65071</a></td> </tr><tr> <td>[4]</td> <td class="space">Efendiev, Y.; Hou, T., Multiscale Finite Element Methods: Theory and Applications, vol. 4, 2009, Springer Science & Business Media · <a href="/1163.65080" class="nowrap">Zbl 1163.65080</a></td> </tr><tr> <td>[5]</td> <td class="space">Efendiev, Y.; Galvis, J.; Hou, T. Y., Generalized multiscale finite element methods (gmsfem), J. Comput. Phys., 251, 116-135, 2013 · <a href="/1349.65617" class="nowrap">Zbl 1349.65617</a></td> </tr><tr> <td>[6]</td> <td class="space">Chung, E.; Efendiev, Y.; Hou, T. Y., Adaptive multiscale model reduction with generalized multiscale finite element methods, J. Comput. Phys., 320, 69-95, 2016 · <a href="/1349.76191" class="nowrap">Zbl 1349.76191</a></td> </tr><tr> <td>[7]</td> <td class="space">Chung, E.; Efendiev, Y.; Hou, T. Y., Multiscale Model Reduction: Multiscale Finite Element Methods and Their Generalizations, vol. 212, 2023, Springer Nature · <a href="/1543.65001" class="nowrap">Zbl 1543.65001</a></td> </tr><tr> <td>[8]</td> <td class="space">Hackbusch, W., Multigrid Methods and Applications, Springer Series in Computational Mathematics, vol. 4, 1985, Springer-Verlag: Springer-Verlag Berlin · <a href="/0577.65118" class="nowrap">Zbl 0577.65118</a></td> </tr><tr> <td>[9]</td> <td class="space">Xu, J.; Zikatanov, L., Algebraic multigrid methods, Acta Numer., 26, 591-721, 2017 · <a href="/1378.65182" class="nowrap">Zbl 1378.65182</a></td> </tr><tr> <td>[10]</td> <td class="space">Brewster, M. E.; Beylkin, G., A multiresolution strategy for numerical homogenization, Appl. Comput. Harmon. 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