Meta-Learning for Physics-Informed Neural Networks (PINNs): A Comprehensive Framework for Few-Shot Adaptation in Parametric PDEs

Meta-Learning for Physics-Informed Neural Networks (PINNs): A Comprehensive Framework for Few-Shot Adaptation in Parametric PDEs | 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 Meta-Learning for Physics-Informed Neural Networks (PINNs): A Comprehensive Framework for Few-Shot Adaptation in Parametric PDEs Brandon Yee, Wilson Collins, Benjamin Pellegrini, Caden Wang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7497594/v2 This work is licensed under a CC BY 4.0 License Status: Posted Version 2 posted You are reading this latest preprint version Show more versions Abstract Physics-Informed Neural Networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by incorporating physical laws directly into neural network training. However, traditional PINNs require extensive retraining for each new PDE configuration, limiting their practical applicability in parametric scenarios. This work presents a comprehensive meta-learning framework for PINNs that enables rapid adaptation to new parametric PDE problems with minimal training data. We introduce four novel meta-learning architectures: MetaPINN, PhysicsInformedMetaLearner, TransferLearningPINN, and DistributedMetaPINN, each designed to address specific challenges in few-shot PDE solving. Through extensive evaluation on seven parametric PDE families including heat equations, Burgers equations, Poisson problems, Navier-Stokes equations, Gray-Scott systems, and Kuramoto-Sivashinsky equations, we demonstrate that meta-learning approaches achieve L2 relative error of 0.034 compared to 0.160 for standard PINNs, representing an 79% error reduction, while reducing adaptation time by 6.5×. Our PhysicsInformedMetaLearner consistently outperforms all baselines across 280 statistical comparisons with 92.9% significance rate. The framework includes comprehensive computational analysis showing break-even points at 13-16 tasks and scalability up to 8 GPUs with 85% parallel efficiency. This work establishes meta-learning as a transformative approach for parametric PDE solving, enabling practical deployment of PINNs in real-time and multi-query scenarios. However, chaotic systems like Kuramoto-Sivashinsky equations present increased challenges, with our best method achieving L2 relative error of 0.089 compared to 0.034 average across all problems, indicating the need for specialized approaches for highly nonlinear dynamics. Artificial Intelligence and Machine Learning Computational Physics meta-learning physics-informed neural networks few-shot learning computational fluid dynamics partial differential equations adaptive constraint weighting automated physics discovery Navier-Stokes equations generalization performance convergence guarantees Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 2 posted You are reading this latest preprint version Show more versions 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-7497594","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":527744723,"identity":"f2299f9f-1992-4151-b2c0-be4ab5dab86c","order_by":0,"name":"Brandon Yee","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA6UlEQVRIiWNgGAWjYBACAzBZASYZDzA2EK3lDIRDghbGNlK0mDPwGD74OO+wvHn/AoYDP3fUMfBLH7+AV4tlA4+x4cxthw3n3HjAcLD3zGEGyb6cAvwOO8BjJs277XCChMT5Dwd42w4wGJzhSSCkxfz33zkgLQcYDv5tqyNKixkzYwNQC38Dw2HeNmagFvYD+P3SzFYs2XMs3XCGBAPDYdkzh3kke3jw6mAwZ2/e+OFHjbW8BP8Bxodvd9TJ8fOwP8Cvh5kDFDXNDAwSEB8AreAxwK+FAWxmHQMD/wEUkVEwCkbBKBgFcAAAlaJIQauKuysAAAAASUVORK5CYII=","orcid":"https://orcid.org/0009-0003-3948-1403","institution":"Yee Collins Research Group","correspondingAuthor":true,"prefix":"","firstName":"Brandon","middleName":"","lastName":"Yee","suffix":""},{"id":527744724,"identity":"ba0221ac-45b2-4600-85d0-77c6bbe6db62","order_by":1,"name":"Wilson Collins","email":"","orcid":"","institution":"Yee Collins Research Group","correspondingAuthor":false,"prefix":"","firstName":"Wilson","middleName":"","lastName":"Collins","suffix":""},{"id":527744725,"identity":"875f2700-647a-4257-bee7-618a3b6cd3fb","order_by":2,"name":"Benjamin Pellegrini","email":"","orcid":"","institution":"Yee Collins Research Group","correspondingAuthor":false,"prefix":"","firstName":"Benjamin","middleName":"","lastName":"Pellegrini","suffix":""},{"id":527744726,"identity":"a848f8e9-acba-4e11-b6ef-a21e698c5345","order_by":3,"name":"Caden Wang","email":"","orcid":"","institution":"New York University","correspondingAuthor":false,"prefix":"","firstName":"Caden","middleName":"","lastName":"Wang","suffix":""}],"badges":[],"createdAt":"2025-08-30 23:04:31","currentVersionCode":2,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-7497594/v2","doiUrl":"https://doi.org/10.21203/rs.3.rs-7497594/v2","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":95799568,"identity":"910407d2-ef9c-49e8-9c2b-bb3ff2bc8a97","added_by":"auto","created_at":"2025-11-13 08:20:17","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2101284,"visible":true,"origin":"","legend":"","description":"","filename":"metapinn42.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7497594/v2_covered_509ec677-cb2f-4eb5-ac1d-e0e8bad24e1a.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eMeta-Learning for Physics-Informed Neural Networks (PINNs): A Comprehensive Framework for Few-Shot Adaptation in Parametric PDEs\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Yee Collins Research Group","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":"meta-learning, physics-informed neural networks, few-shot learning, computational fluid dynamics, partial differential equations, adaptive constraint weighting, automated physics discovery, Navier-Stokes equations, generalization performance, convergence guarantees","lastPublishedDoi":"10.21203/rs.3.rs-7497594/v2","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7497594/v2","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003ePhysics-Informed Neural Networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by incorporating physical laws directly into neural network training. 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