GDEA: Global-Weight Deep Equilibrium Attention for Finite Element Systems

GDEA: Global-Weight Deep Equilibrium Attention for Finite Element Systems | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article GDEA: Global-Weight Deep Equilibrium Attention for Finite Element Systems Junghun Lee, Conrad Tucker This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9316051/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted You are reading this latest preprint version Abstract Graph neural networks have been explored as surrogate models for the finite element method due to the computational cost. However, many models fail to effectively capture the influence of distant nodes. This is critical in surrogate models because boundary conditions at a distance can significantly influence the results. To mitigate these issues, we propose iterative message passing using a global-weight matrix assembled via the direct stiffness method. This method is implemented through a deep equilibrium model with Anderson acceleration to ensure fast convergence and low memory cost. Our model is evaluated on published FEM datasets as well as a new dataset, Deformed ABC, featuring diverse geometries, materials, and boundary conditions. Our model outperforms others across all criteria, including a 12% lower average RMSE and a 5% higher average R 2 on a complex new dataset, but differences are marginal due to saturation of the benchmark across all models in simpler datasets. Physical sciences/Engineering/Mechanical engineering Physical sciences/Mathematics and computing/Computational science Surrogate model Graph Neural Network Finite element method Full Text Additional Declarations There is NO Competing Interest. Supplementary Files GDEAGlobalWeightDeepEquilibriumAttentionSupplementaryMaterials.pdf GDEA Global-Weight Deep Equilibrium Attention_Supplementary Materials Cite Share Download PDF Status: Under Review Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9316051","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":622383879,"identity":"69411c20-2767-4fb5-a1ea-33c93e756bcf","order_by":0,"name":"Junghun Lee","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA8UlEQVRIiWNgGAWjYBACxgYGBmYgLcMPJCQQ4myEtfBINhCrBQTAWgwOEKuFeUbu4deFbXd4jI+fPXi7ouIwA/+0MwYMH8oO43bYjLw065ltz3jMzuQlW545c5hB4naOAeOMc/i05JgZ87Yd5jE7kGMm2dh2m8FAOseAGShCWItx/xugln9QLX/xazF+DNJiIAGypQGqhRGflp43Zsw85w7zSNx4Y2zZcOw/j8TttIKDPefScWoxbM8x/sxTdliOvz/H8GZDTZoc/+zkjQ9+lFnj1tLAwCaBLMADIg7gVA8E8sCo+YBPwSgYBaNgFIwCBgCiH1J+LlUdjQAAAABJRU5ErkJggg==","orcid":"","institution":"Carnegie Mellon University","correspondingAuthor":true,"prefix":"","firstName":"Junghun","middleName":"","lastName":"Lee","suffix":""},{"id":622383880,"identity":"9f272092-0db9-45f5-a036-c6ed7f2cf7bb","order_by":1,"name":"Conrad Tucker","email":"","orcid":"","institution":"Carnegie Mellon University","correspondingAuthor":false,"prefix":"","firstName":"Conrad","middleName":"","lastName":"Tucker","suffix":""}],"badges":[],"createdAt":"2026-04-03 20:30:36","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9316051/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9316051/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":106960701,"identity":"78ad8e9d-b2cd-4330-ab36-1aab9d1f01ad","added_by":"auto","created_at":"2026-04-15 09:22:43","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":873764,"visible":true,"origin":"","legend":"Article File","description":"","filename":"GDEAGlobalWeightDeepEquilibriumAttention.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9316051/v1_covered_18665f16-211f-42df-94ef-b01ce59dd65c.pdf"},{"id":106886922,"identity":"6cf76528-3b99-4246-b21d-ce669f5f4eaa","added_by":"auto","created_at":"2026-04-14 12:45:16","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":474642,"visible":true,"origin":"","legend":"GDEA Global-Weight Deep Equilibrium Attention_Supplementary Materials","description":"","filename":"GDEAGlobalWeightDeepEquilibriumAttentionSupplementaryMaterials.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9316051/v1/52b2a4144bb33840a91bb316.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"GDEA: Global-Weight Deep Equilibrium Attention for Finite Element Systems","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nature-portfolio","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"","title":"Nature Portfolio","twitterHandle":"","acdcEnabled":false,"dfaEnabled":false,"editorialSystem":"ejp","reportingPortfolio":"","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Surrogate model, Graph Neural Network, Finite element method","lastPublishedDoi":"10.21203/rs.3.rs-9316051/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9316051/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Graph neural networks have been explored as surrogate models for the finite element method due to the computational cost. 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