Reimagining Fisher’s Equation via Physics-Informed Neural Networks

Reimagining Fisher’s Equation via Physics-Informed Neural Networks | 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 Reimagining Fisher’s Equation via Physics-Informed Neural Networks Gaurav Kumar, Mahima Lakra, Sumit Malik, Sanjay Kumar This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7953742/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Purpose: We implement a modern Physics-Informed Neural Networks (PINNs) algorithm that markedly enhances the computational execution of classical and generalized Fisher equations by easily integrating physical governing laws into supervised learning structures. \textbf{Methods:} We are using the Physics-Informed Neural Network framework to minimize the loss function using different optimization algorithms. This improves the accuracy of complex spatio-temporal dynamics by combining data-driven optimization with conservation principles. Results: Rigorous validation across multiple Fisher equation variants exhibits computational precision, with optimum estimates that achieve error magnitudes of 10 -6 while demonstrating outstanding numerical stability and convergence features. A full performance analysis shows that our method is better at predicting than existing finite-difference and spectral methods, especially in areas with steep gradients and nonlinear transition zones. Conclusion: We deploy a mesh-independent Physics-Informed Neural Network (PINN) framework that overcomes the limits of existing solvers, assuring computational efficiency and scalability. The technique yields correct solutions for complex reaction–diffusion systems and demonstrates substantial potential for simulating biological and physical phenomena, including pattern creation and population dynamics. By eliminating the necessity for mesh-based methods, this framework offers a viable alternative for solving nonlinear PDEs, especially in tough cases where standard approaches fall short. physics-informed neural networks Fisher equation reaction–diffusion modeling mesh-free methods Deep learning Partial differential equations Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted 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-7953742","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":536629899,"identity":"30e2f643-80e1-4ab3-abf0-823ec1e0ace8","order_by":0,"name":"Gaurav Kumar","email":"","orcid":"","institution":"School of Computer Science and Engineering and Technology (SCSET), Bennett University","correspondingAuthor":false,"prefix":"","firstName":"Gaurav","middleName":"","lastName":"Kumar","suffix":""},{"id":536629900,"identity":"c9a90406-60c1-49ac-b33f-ba94ce9e45b5","order_by":1,"name":"Mahima Lakra","email":"","orcid":"","institution":"National 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Networks\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"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":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"physics-informed neural networks, Fisher equation, reaction–diffusion modeling, mesh-free 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This improves the accuracy of complex spatio-temporal dynamics by combining data-driven optimization with conservation principles.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eResults:\u003c/strong\u003e Rigorous validation across multiple Fisher equation variants exhibits computational precision, with optimum estimates that achieve error magnitudes of 10\u003csup\u003e-6\u003c/sup\u003e while demonstrating outstanding numerical stability and convergence features. A full performance analysis shows that our method is better at predicting than existing finite-difference and spectral methods, especially in areas with steep gradients and nonlinear transition zones.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConclusion:\u003c/strong\u003e We deploy a mesh-independent Physics-Informed Neural Network (PINN) framework that overcomes the limits of existing solvers, assuring computational efficiency and scalability. The technique yields correct solutions for complex reaction–diffusion systems and demonstrates substantial potential for simulating biological and physical phenomena, including pattern creation and population dynamics. 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