Hybrid FEM–PINN Framework for 2D Shallow-Water Flow Simulation and Error Analysis

Hybrid FEM–PINN Framework for 2D Shallow-Water Flow Simulation and Error Analysis | 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 Research Article Hybrid FEM–PINN Framework for 2D Shallow-Water Flow Simulation and Error Analysis Hamid Ebrahimi, Atefe Morovati This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7796717/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 5 You are reading this latest preprint version Abstract Accurate and efficient simulation of shallow water dynamics remains one of the most challenging tasks in computational hydraulics due to the inherent trade-off between physical conservation and numerical stability. Conventional finite element methods (FEM) are known for their strong adherence to conservation principles, particularly mass balance, yet they frequently suffer from excessive diffusion, oscillations, and limited adaptability when dealing with highly nonlinear or data-sparse environments. In contrast, physics-informed neural networks (PINNs) have emerged as powerful tools that embed governing equations into the training of deep learning models, thereby providing smooth, data-driven solutions even in the absence of dense measurements. However, the standalone application of PINNs is restricted by weak enforcement of physical invariants, accumulation of errors over long-time horizons, and difficulties in achieving robust convergence. This methodological gap highlights the urgent need for a hybrid framework that can combine the complementary strengths of FEM and PINNs. The present study introduces an innovative hybrid FEM+PINN framework designed to close this gap. FEM is first employed to generate high-fidelity snapshots of the shallow water system, which serve as structured training data to guide the PINN component. The PINN simultaneously incorporates the shallow water governing equations, conservation constraints, and boundary conditions, ensuring that physical laws are respected during learning. By merging the conservation properties of FEM with the flexibility and adaptivity of PINNs, the proposed model achieves a balance between accuracy, robustness, and generalization. The training strategy further implements collocation sampling, gradient clipping, and dynamic weighting of data and PDE losses to enhance stability across epochs. The results confirm the clear advantages of the hybrid approach. Quantitatively, the water depth h achieves a root mean square error (RMSE) of 0.0461 and mean absolute error (MAE) of 0.0403, while the momentum components hu and hv exhibit RMSE values of 0.1265 and 0.1, respectively. Mass conservation is preserved with deviations limited to 3.55%, whereas energy conservation, although not exact, demonstrates controlled systematic accumulation rather than instability. The training loss decreases rapidly from an initial value of 6.0×105 to below 2.0×104 within 60 epochs, confirming convergence without oscillatory artifacts. Spatial error distributions remain smooth and symmetric, error histograms demonstrate strong clustering near zero, and flow field visualizations highlight coherent structures without noise. This study fills a crucial methodological gap by demonstrating that neither FEM nor PINNs alone can fully meet the requirements of accuracy, conservation, and stability in shallow water modeling. The hybrid FEM+PINN framework provides a novel, physically consistent, and computationally efficient solution. Its ability to integrate classical numerical rigor with machine learning adaptability marks a significant advancement for data-driven hydraulics and paves the way for next-generation models in environmental and engineering fluid dynamics. Shallow Water Equations Physics-Informed Neural Networks (PINN) Finite Element Method (FEM) Hybrid Modeling Data-Driven Simulation Hydrodynamics Full Text Cite Share Download PDF Status: Under Review Version 1 posted Editor invited by journal 26 Feb, 2026 Reviewers agreed at journal 08 Oct, 2025 Reviewers invited by journal 08 Oct, 2025 Editor assigned by journal 08 Oct, 2025 First submitted to journal 07 Oct, 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-7796717","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":526599524,"identity":"8bdb894f-ad6e-46fe-bac1-28335a66ebaf","order_by":0,"name":"Hamid 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Analysis","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"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":"water-resources-management","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"warm","sideBox":"Learn more about [Water Resources Management](https://www.springer.com/journal/11269)","snPcode":"11269","submissionUrl":"https://submission.nature.com/new-submission/11269/3","title":"Water Resources Management","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Shallow Water Equations, Physics-Informed Neural Networks (PINN), Finite Element Method (FEM), Hybrid Modeling, Data-Driven Simulation, Hydrodynamics","lastPublishedDoi":"10.21203/rs.3.rs-7796717/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7796717/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAccurate and efficient simulation of shallow water dynamics remains one of the most challenging tasks in computational hydraulics due to the inherent trade-off between physical conservation and numerical stability. Conventional finite element methods (FEM) are known for their strong adherence to conservation principles, particularly mass balance, yet they frequently suffer from excessive diffusion, oscillations, and limited adaptability when dealing with highly nonlinear or data-sparse environments. In contrast, physics-informed neural networks (PINNs) have emerged as powerful tools that embed governing equations into the training of deep learning models, thereby providing smooth, data-driven solutions even in the absence of dense measurements. However, the standalone application of PINNs is restricted by weak enforcement of physical invariants, accumulation of errors over long-time horizons, and difficulties in achieving robust convergence. This methodological gap highlights the urgent need for a hybrid framework that can combine the complementary strengths of FEM and PINNs. The present study introduces an innovative hybrid FEM+PINN framework designed to close this gap. FEM is first employed to generate high-fidelity snapshots of the shallow water system, which serve as structured training data to guide the PINN component. The PINN simultaneously incorporates the shallow water governing equations, conservation constraints, and boundary conditions, ensuring that physical laws are respected during learning. By merging the conservation properties of FEM with the flexibility and adaptivity of PINNs, the proposed model achieves a balance between accuracy, robustness, and generalization. The training strategy further implements collocation sampling, gradient clipping, and dynamic weighting of data and PDE losses to enhance stability across epochs. The results confirm the clear advantages of the hybrid approach. Quantitatively, the water depth h achieves a root mean square error (RMSE) of 0.0461 and mean absolute error (MAE) of 0.0403, while the momentum components hu and hv exhibit RMSE values of 0.1265 and 0.1, respectively. Mass conservation is preserved with deviations limited to 3.55%, whereas energy conservation, although not exact, demonstrates controlled systematic accumulation rather than instability. The training loss decreases rapidly from an initial value of 6.0×105 to below 2.0×104 within 60 epochs, confirming convergence without oscillatory artifacts. Spatial error distributions remain smooth and symmetric, error histograms demonstrate strong clustering near zero, and flow field visualizations highlight coherent structures without noise. This study fills a crucial methodological gap by demonstrating that neither FEM nor PINNs alone can fully meet the requirements of accuracy, conservation, and stability in shallow water modeling. The hybrid FEM+PINN framework provides a novel, physically consistent, and computationally efficient solution. Its ability to integrate classical numerical rigor with machine learning adaptability marks a significant advancement for data-driven hydraulics and paves the way for next-generation models in environmental and engineering fluid dynamics.\u003c/p\u003e","manuscriptTitle":"Hybrid FEM–PINN Framework for 2D Shallow-Water Flow Simulation and Error Analysis","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-10-21 03:32:42","doi":"10.21203/rs.3.rs-7796717/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"editorInvited","content":"Water Resources Management","date":"2026-02-26T14:51:11+00:00","index":"","fulltext":""},{"type":"reviewerAgreed","content":"","date":"2025-10-08T14:57:06+00:00","index":0,"fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-10-08T14:37:47+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-10-08T12:53:15+00:00","index":"","fulltext":""},{"type":"submitted","content":"Water Resources Management","date":"2025-10-07T14:41:28+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"water-resources-management","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"warm","sideBox":"Learn more about [Water Resources Management](https://www.springer.com/journal/11269)","snPcode":"11269","submissionUrl":"https://submission.nature.com/new-submission/11269/3","title":"Water Resources Management","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"255cb3fe-e8d8-4d35-8204-e69279925894","owner":[],"postedDate":"October 21st, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2025-10-21T03:32:42+00:00","versionOfRecord":[],"versionCreatedAt":"2025-10-21 03:32:42","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7796717","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7796717","identity":"rs-7796717","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}