NPINN+: An enhanced physics-informed neural network for solving wave equations with nonlocal boundary conditions

NPINN+: An enhanced physics-informed neural network for solving wave equations with nonlocal boundary conditions | 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 NPINN+: An enhanced physics-informed neural network for solving wave equations with nonlocal boundary conditions Qiancheng Tan, Shuyun Yang, Yonghui Qin This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8884832/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 12 You are reading this latest preprint version Abstract Wave equations with nonlocal conditions appear in many scientic and engineering applications, such as, the population dynamics, the mathematical biology, and the materials science. The numerical challenge mainly stems from nonlocal terms, whose global coupling degrades the efficiency and stability of classical methods. In recent years, physics-informed neural networks (PINN) have achieved notable success in solving partial differential equations.In this paper, we propose an enhanced physics-informed neural network for wave equations subject to nonlocal conditions, termed NPINN+. By exploiting an equivalent transformation of the nonlocal condition, the original problem is reformulated into a wave equation satisfying Neumann boundary conditions with an additional integral-form source term. NPINN+ employs a single neural network to provide a unified representation of the spatiotemporal solution, while incorporating the governing equation, derivative information, initial and boundary conditions, and nonlocal constraints into a unified physics-informed loss function, enabling effective capture of the underlying physical features.Furthermore, a residual-based dynamic sampling strategy and a SoftAdapt-driven adaptive loss weighting mechanism are introduced to enhance accuracy and training robustness. Numerical experiments on regular domains demonstrate the effectiveness of the proposed method, and its extension to star-shaped domains is achieved via a polar coordinate transformation. Comparative results with PINN, APINN, and RAR-PINN show that NPINN+ consistently achieves superior accuracy and stability. Physical sciences/Mathematics and computing Physical sciences/Physics Physics-informed neural networks (PINN) Nonlocal condition Wave equation Numerical simulation Dynamic softAdapt loss weighting Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 16 Mar, 2026 Reviews received at journal 15 Mar, 2026 Reviews received at journal 08 Mar, 2026 Reviewers agreed at journal 04 Mar, 2026 Reviewers agreed at journal 02 Mar, 2026 Reviews received at journal 27 Feb, 2026 Reviewers agreed at journal 25 Feb, 2026 Reviewers invited by journal 25 Feb, 2026 Editor assigned by journal 25 Feb, 2026 Editor invited by journal 25 Feb, 2026 Submission checks completed at journal 22 Feb, 2026 First submitted to journal 22 Feb, 2026 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-8884832","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":598613843,"identity":"3bae137c-7e7e-483b-b172-f8b7600945b4","order_by":0,"name":"Qiancheng Tan","email":"","orcid":"","institution":"Guilin University of Electronic Technology","correspondingAuthor":false,"prefix":"","firstName":"Qiancheng","middleName":"","lastName":"Tan","suffix":""},{"id":598613844,"identity":"956fb442-1bde-4b68-aed4-c98033cacf95","order_by":1,"name":"Shuyun Yang","email":"","orcid":"","institution":"Guilin University of Electronic Technology","correspondingAuthor":false,"prefix":"","firstName":"Shuyun","middleName":"","lastName":"Yang","suffix":""},{"id":598613845,"identity":"adea086d-590a-4224-a70c-802de4d6e174","order_by":2,"name":"Yonghui Qin","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3ElEQVRIiWNgGAWjYJACZgYGCR5++ccHHyRU1BCvRUayIS3Z4MGZY0RrYbAxaMhRk3zYwkxYucHx3sOvC9sseAwYzrBVJDawMfC3dyfg13LmXJr1zDYJHnPG3mM3EnfIMEicObsBv5YbOWbGvEAtls18aTcSz7AxGEjkEqnF4BiPWUFiGzNRWowfg7Wc4TFjIEqL5JkzZsw85yR4JGewJUsknDnGQ9AvfMd7jD/zlNXZ80swH/z4o6JGjr+9F78WhQMMbBLIAjx4lYOAfAMD8weCqkbBKBgFo2BkAwAmX0aC/he6QgAAAABJRU5ErkJggg==","orcid":"","institution":"Guilin University of Electronic Technology","correspondingAuthor":true,"prefix":"","firstName":"Yonghui","middleName":"","lastName":"Qin","suffix":""}],"badges":[],"createdAt":"2026-02-15 08:54:11","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8884832/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8884832/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104400328,"identity":"2b7dc400-e94d-4905-9007-29fccc8e1f1b","added_by":"auto","created_at":"2026-03-11 12:09:39","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":27567309,"visible":true,"origin":"","legend":"","description":"","filename":"NPINN3DWavesubmitR20.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8884832/v1_covered_b9aa9c0e-0f1d-4d4b-b7b6-0b9e9c85a4c0.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eNPINN+: An enhanced physics-informed neural network for solving wave equations with nonlocal boundary conditions\u003c/p\u003e","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":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Physics-informed neural networks (PINN), Nonlocal condition, Wave equation, Numerical simulation, Dynamic softAdapt loss weighting","lastPublishedDoi":"10.21203/rs.3.rs-8884832/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8884832/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWave equations with nonlocal conditions appear in many scientic and engineering applications, such as, the population dynamics, the mathematical biology, and the materials science. The numerical challenge mainly stems from nonlocal terms, whose global coupling degrades the efficiency and stability of classical methods. In recent years, physics-informed neural networks (PINN) have achieved notable success in solving partial differential equations.In this paper, we propose an enhanced physics-informed neural network for wave equations subject to nonlocal conditions, termed NPINN+. By exploiting an equivalent transformation of the nonlocal condition, the original problem is reformulated into a wave equation satisfying Neumann boundary conditions with an additional integral-form source term. 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