From mesh to neural nets: A multi-method evaluation of physics informed neural network and galerkin finite element method for solving nonlinear convection-reaction-diffusion equations

From mesh to neural nets: A multi-method evaluation of physics informed neural network and galerkin finite element method for solving nonlinear convection-reaction-diffusion equations | 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 From mesh to neural nets: A multi-method evaluation of physics informed neural network and galerkin finite element method for solving nonlinear convection-reaction-diffusion equations Fardous Hasan, Hazrat Ali, Hasan Asyari Arief This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5447948/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 8 You are reading this latest preprint version Abstract Non-linear convection-reaction-diffusion (CRD) partial differential equations (PDEs) are crucial for modeling complex phenomena in fields such as biology, ecology, population dynamics, physics, and engineering. Numerical approximation of these non-linear systems is essential due to the challenges of obtaining exact solutions. Traditionally, the Galerkin finite element method (GFEM) has been the standard computational tool for solving these PDEs. With the advancements in machine learning, Physics-Informed Neural Network (PINN) has emerged as a promising alternative for approximating non-linear PDEs.In this study, we compare the performance of PINN and GFEM by solving four distinct one-dimensional CRD problems with varying initial and boundary conditions and evaluate the performance of PINN over GFEM. This evaluation metrics includes error estimates, and visual representations of the solutions, supported by statistical methods such as the root mean squared error (RMSE), the standard deviation of error, the the Wilcoxon Signed-Rank Test and the coefficient of variation (CV) test.Our findings reveal that while both methods achieve solutions close to the analytical results, PINN demonstrate superior accuracy and efficiency. PINN achieved significantly lower RMSE values and smaller standard deviations for Burgers' equation, Fisher's equation, and Newell-Whitehead-Segel equation, indicating higher accuracy and greater consistency. While GFEM shows slightly better accuracy for the Burgers-Huxley equation, its performance was less consistent over time. In contrast, PINN exhibit more reliable and robust performance, highlighting their potential as a cutting-edge approach for solving non-linear PDEs. Nonlinear Partial differential equation Convection reaction diffusion Physics Informed Neural Network Galerkin finite element method Machine learning Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 07 Mar, 2025 Reviewers agreed at journal 25 Dec, 2024 Reviews received at journal 23 Dec, 2024 Reviewers agreed at journal 18 Dec, 2024 Reviewers invited by journal 13 Dec, 2024 Editor assigned by journal 29 Nov, 2024 Submission checks completed at journal 14 Nov, 2024 First submitted to journal 13 Nov, 2024 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-5447948","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":384463781,"identity":"4f1d6102-0f8e-44bf-b4b3-a2645be9c9d7","order_by":0,"name":"Fardous Hasan","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA10lEQVRIiWNgGAWjYBAC+wMMDBIghsEBEFnBwGNASAtIAZKWMyRrYWyDiODXwn724Y2PbXUMBsePP/xcOO+wjDkD78EH+LTY86QbW85sO8xgfybHWHrmtsM8lg18yXhtMmBIY5PmbTsAdFgOgzQvUIvBAR4zCbxa+J+BtAAddv7549+8c4jRIgG2hZnB4EaCmTRvA1FanjFbzjgHVHnjjZk1z7F0HstmHmP8fuFPY7zxoaxOzuB8+uPbPDXW9ubsPYYP8GmBAR4Ek5kY9aNgFIyCUTAK8AIAgs1CQZd1pVcAAAAASUVORK5CYII=","orcid":"","institution":"University of Bergen","correspondingAuthor":true,"prefix":"","firstName":"Fardous","middleName":"","lastName":"Hasan","suffix":""},{"id":384463782,"identity":"0d9b5440-02da-42bd-b420-8033a981f97f","order_by":1,"name":"Hazrat Ali","email":"","orcid":"","institution":"The University of Texas at El Paso","correspondingAuthor":false,"prefix":"","firstName":"Hazrat","middleName":"","lastName":"Ali","suffix":""},{"id":384463783,"identity":"8f1af62f-b7c8-4e0b-9639-b7b33dc557bd","order_by":2,"name":"Hasan Asyari Arief","email":"","orcid":"","institution":"NORCE Norwegian Research Centre AS","correspondingAuthor":false,"prefix":"","firstName":"Hasan","middleName":"Asyari","lastName":"Arief","suffix":""}],"badges":[],"createdAt":"2024-11-13 14:53:25","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5447948/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5447948/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":70303410,"identity":"0d0ee827-e62e-465a-ae8e-8b25214b5f56","added_by":"auto","created_at":"2024-12-02 02:35:34","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1194494,"visible":true,"origin":"","legend":"","description":"","filename":"reviseddocuments.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5447948/v1_covered_4cdf8d22-d66b-4d6a-9be2-efe039f0b863.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"From mesh to neural nets: A multi-method evaluation of physics informed neural network and galerkin finite element method for solving nonlinear convection-reaction-diffusion equations","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"international-journal-of-applied-and-computational-mathematics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [International Journal of Applied and Computational Mathematics](https://link.springer.com/journal/40819)","snPcode":"40819","submissionUrl":"https://submission.nature.com/new-submission/40819/3","title":"International Journal of Applied and Computational Mathematics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Nonlinear Partial differential equation, Convection reaction diffusion, Physics Informed Neural Network, Galerkin finite element method, Machine learning","lastPublishedDoi":"10.21203/rs.3.rs-5447948/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5447948/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eNon-linear convection-reaction-diffusion (CRD) partial differential equations (PDEs) are crucial for modeling complex phenomena in fields such as biology, ecology, population dynamics, physics, and engineering. 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