Physics-informed Fourier Basis Neural Network for Fluid Mechanics

Physics-informed Fourier Basis Neural Network for Fluid Mechanics | 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 Physics-informed Fourier Basis Neural Network for Fluid Mechanics Chao Wang, Shilong Li, Zelong Yuan, Chunyu Guo This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7693532/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 Conventional machine learning approaches struggle to capture periodic patterns and solve quasi-periodic boundary problems in fluid mechanics. This study proposes a physics-informed Fourier basis neural network (PIFBNN) that integrates adaptive Fourier series with physical constraints to address canonical partial differential equations. The architecture preserves Fourier series' natural mathematical compatibility with periodic phenomena while incorporating trainable parameters (angular frequencies and weight coefficients) that enhance basis function flexibility and nonlinear learning capacity. We evaluate the framework on six fundamental fluid dynamics benchmarks: two-dimensional cylinder wake flow, lid-driven cavity flow, Kovasznay flow, Helmholtz equation, Burgers equation, and Allen-Cahn equation. Results demonstrate PIFBNN's consistent superiority over standard physics-informed neural network (PINN) in accuracy. Through sparse data reconstruction experiments and adjusting the activation functions of neural networks and comparing , we further validate the dual advantages of Fourier basis neural network (FBNN) over conventional artificial neural network (ANN): inherent periodicity handling and reduced sensitivity to activation function selection. The FBNN architecture maintains robust performance across different activation functions, as verified through systematic comparisons with ANN and PINN baselines. These findings position PIFBNN as a promising computational framework for complex fluid dynamics problems. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 20 Apr, 2026 Reviewers invited by journal 16 Oct, 2025 Editor assigned by journal 24 Sep, 2025 Submission checks completed at journal 24 Sep, 2025 First submitted to journal 23 Sep, 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. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-7693532","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":535079904,"identity":"e6ec4ad7-5d01-429c-8ba9-27bdc4cbc09e","order_by":0,"name":"Chao Wang","email":"","orcid":"","institution":"Harbin Engineering University","correspondingAuthor":false,"prefix":"","firstName":"Chao","middleName":"","lastName":"Wang","suffix":""},{"id":535079906,"identity":"236e5cf8-fd12-44cf-8ade-d8eec69940ed","order_by":1,"name":"Shilong Li","email":"","orcid":"","institution":"Harbin Engineering University","correspondingAuthor":false,"prefix":"","firstName":"Shilong","middleName":"","lastName":"Li","suffix":""},{"id":535079907,"identity":"fe9b9f0e-70b9-45f2-84b1-dac80b12dd51","order_by":2,"name":"Zelong Yuan","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+ElEQVRIiWNgGAWjYPACCyBmPsbA2ADmGRCjRQKI2dJI1sJjRpwWgxvJzx7+qJCQM+df8+1x5Q6bxAb25m0SDDV38GhJMzeQOCNhbDnj7XbDs2fSEht4jpVJMBx7hkdLgpmEYZtE4oYbZ7dJNrYdTmyQyDGTYGw4jEdL+jeJxH8gLWeeAbX8T2yQf0NIC9DMgw1ALed72IBaDgBt4cGvRfLMmzLJhmMSxgY32MwkG88kG7fxpBVbJBzDrYXvePo2yR81NnIG5w8DHbbDTraf/fDGGx9qcGtROABjSSRAaDYQkYBTAwODfAOMxX8At6pRMApGwSgY2QAAAZZZTYps9cgAAAAASUVORK5CYII=","orcid":"","institution":"Harbin Engineering University","correspondingAuthor":true,"prefix":"","firstName":"Zelong","middleName":"","lastName":"Yuan","suffix":""},{"id":535079908,"identity":"b98d49c3-3049-4611-a544-548a9044ca6a","order_by":3,"name":"Chunyu Guo","email":"","orcid":"","institution":"Harbin Engineering University","correspondingAuthor":false,"prefix":"","firstName":"Chunyu","middleName":"","lastName":"Guo","suffix":""}],"badges":[],"createdAt":"2025-09-23 11:08:19","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7693532/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7693532/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":94717239,"identity":"323ae20f-7c0b-4bac-9f11-113c5f9c667e","added_by":"auto","created_at":"2025-10-30 03:56:54","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":5961,"visible":true,"origin":"","legend":"","description":"","filename":"353b509d3dbd477ab275856070b27498.json","url":"https://assets-eu.researchsquare.com/files/rs-7693532/v1/6b52aa9b830eadb6d104cf9f.json"},{"id":94730034,"identity":"b6dbaff1-853f-44aa-b970-4e900cff2c81","added_by":"auto","created_at":"2025-10-30 07:05:36","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":14608950,"visible":true,"origin":"","legend":"","description":"","filename":"manuscriptEC.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7693532/v1_covered_49e95c03-3648-4177-bca6-9b61ca49dbc3.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Physics-informed Fourier Basis Neural Network for Fluid Mechanics","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":"engineering-with-computers","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"ewco","sideBox":"Learn more about [Engineering with Computers](http://link.springer.com/journal/366)","snPcode":"366","submissionUrl":"https://submission.nature.com/new-submission/366/3","title":"Engineering with Computers","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-7693532/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7693532/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Conventional machine learning approaches struggle to capture periodic patterns and solve quasi-periodic boundary problems in fluid mechanics. 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