The Finite Element Neural Network Method: Leveraging Non-Vanishing Shape Functions in Space-Time-Parameter Framework

The Finite Element Neural Network Method: Leveraging Non-Vanishing Shape Functions in Space-Time-Parameter Framework | 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 The Finite Element Neural Network Method: Leveraging Non-Vanishing Shape Functions in Space-Time-Parameter Framework Mohammed Abda, Lucas Berthet, Mohsen Hamedi, Dani Hibatullah, and 4 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7702801/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 18 Apr, 2026 Read the published version in Scientific Reports → Version 1 posted 11 You are reading this latest preprint version Abstract Neural networks (NNs) have received growing interest in engineering due to their ability to assimilate high-dimensional data and provide accurate approximations for complex systems. Nonetheless, classical numerical methods remain the benchmark for reliability and accuracy, backed by rigorous development over decades. Merging the strengths of the finite element formulation with physics-informed neural networks (PINNs), the finite element neural network method (FENNM) opens new venues for approximating partial differential equations (PDEs). FENNM is based on the Petrov-Galerkin framework, where the NN provides the global nonlinear space of solutions, whereas the test functions are the nonvanishing Lagrange shape functions. Compared to VPINN, hp-VPINN, cv-PINN, and FastVPINN, FENNM’s weak-form explicitly includes flux terms at the elements' interfaces and naturally incorporates Neumann boundary conditions within the residual loss function, improving the training stability and adaptability to real-world applications. We extend FENNM to two-dimensional domains, with the second dimension representing space, time, or a parameter. The method naturally integrates time and parameter spaces for design optimization, offering advantages over the deep energy method (DEM) and discrete finite element method (FEM) inspired NNs. We further showcase FENNM’s capability in local mesh refinement, vector-valued PDEs, inverse problems, and complex geometries with irregular elements. Physical sciences/Engineering Physical sciences/Mathematics and computing Physical sciences/Physics Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 18 Apr, 2026 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 24 Nov, 2025 Reviews received at journal 20 Nov, 2025 Reviews received at journal 30 Oct, 2025 Reviewers agreed at journal 03 Oct, 2025 Reviewers agreed at journal 02 Oct, 2025 Reviewers agreed at journal 01 Oct, 2025 Reviewers invited by journal 01 Oct, 2025 Editor invited by journal 01 Oct, 2025 Editor assigned by journal 26 Sep, 2025 Submission checks completed at journal 25 Sep, 2025 First submitted to journal 24 Sep, 2025 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. 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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-7702801","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":524663718,"identity":"599afd4f-6467-4c2a-9434-d03146589060","order_by":0,"name":"Mohammed Abda","email":"","orcid":"","institution":"Polytechnique Montréal","correspondingAuthor":false,"prefix":"","firstName":"Mohammed","middleName":"","lastName":"Abda","suffix":""},{"id":524663719,"identity":"c9515e70-8a50-450a-8cbc-e18c2709d77a","order_by":1,"name":"Lucas Berthet","email":"","orcid":"","institution":"Polytechnique Montréal","correspondingAuthor":false,"prefix":"","firstName":"Lucas","middleName":"","lastName":"Berthet","suffix":""},{"id":524663720,"identity":"3587331e-595d-45db-b5a2-c89c79e1c123","order_by":2,"name":"Mohsen Hamedi","email":"","orcid":"","institution":"Polytechnique Montréal","correspondingAuthor":false,"prefix":"","firstName":"Mohsen","middleName":"","lastName":"Hamedi","suffix":""},{"id":524663721,"identity":"431148e0-bf2c-4cac-90f3-aa633b151a18","order_by":3,"name":"Dani Hibatullah","email":"","orcid":"","institution":"Polytechnique Montréal","correspondingAuthor":false,"prefix":"","firstName":"Dani","middleName":"","lastName":"Hibatullah","suffix":""},{"id":524663722,"identity":"81d8b044-c110-4ca4-ac48-e1ba8ff6bb26","order_by":4,"name":"Elsa Piollet","email":"","orcid":"","institution":"Maya HTT","correspondingAuthor":false,"prefix":"","firstName":"Elsa","middleName":"","lastName":"Piollet","suffix":""},{"id":524663723,"identity":"5aa56aee-3347-49b3-9c5f-9bce7691ba7a","order_by":5,"name":"Christopher Blake","email":"","orcid":"","institution":"Maya HTT","correspondingAuthor":false,"prefix":"","firstName":"Christopher","middleName":"","lastName":"Blake","suffix":""},{"id":524663724,"identity":"3c3a2ab1-3c93-4a63-8439-3705dcd1cd02","order_by":6,"name":"Bruno Blais","email":"","orcid":"","institution":"Polytechnique Montréal","correspondingAuthor":false,"prefix":"","firstName":"Bruno","middleName":"","lastName":"Blais","suffix":""},{"id":524663725,"identity":"89e9e765-7930-44ae-8fd2-f27fa6dbc6f7","order_by":7,"name":"Frédérick P. 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