A Mesh-Free Physics-Informed Neural Network Based on Rayleigh Quotient for Structural Frequency Analysis of Beams and Plates

A Mesh-Free Physics-Informed Neural Network Based on Rayleigh Quotient for Structural Frequency Analysis of Beams and Plates | 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 A Mesh-Free Physics-Informed Neural Network Based on Rayleigh Quotient for Structural Frequency Analysis of Beams and Plates Krishna Kant Mishra, Rajnish Mallick, DineshKumar Harursampath This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7088470/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 Accurate estimation of natural frequencies is essential in structural engineering applications involving beams and plates. Traditional finite element methods are computationally expensive, especially for complex geometries. This paper presents an unsupervised, mesh-free framework based on Physics-Informed Neural Networks (PINNs) that leverages the Rayleigh quotient as a variational loss functional. The proposed method models the mode shape using neural networks while minimizing the ratio of total strain energy to kinetic energy. The approach accurately predicts the fundamental frequencies of Euler-Bernoulli beams and Kirchhoff--Love plates under various classical boundary conditions. Benchmark comparisons with analytical solutions show relative errors as low as $10^{-5}$. The method converges rapidly, requiring fewer collocation points and less training time. This is the first unsupervised PINN framework based on Rayleigh quotient to achieve near-zero error in fundamental frequency extraction for both 1D and 2D structural domains. The method generalizes across different boundary conditions and geometries and offers a scalable alternative to traditional solvers. Mechanical Engineering Artificial Intelligence and Machine Learning Numerical Analysis Physics-Informed Neural Networks (PINNs) Rayleigh Quotient Structural Frequency Estimation Euler-Bernoulli Beams Kirchhoff-Love Plates Deep Learning in Mechanics Full Text Additional Declarations The authors declare no competing interests. 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-7088470","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":483246503,"identity":"197fc7c1-346e-474b-b53a-8d83d28bb3c3","order_by":0,"name":"Krishna Kant Mishra","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3ElEQVRIiWNgGAWjYBACCQY2BoYEhgMMDOyNDQcgYowNRGrhOUiKFgaQFokEIh0mOe1Y2ocHf+7k8c983HjwRw2DPH8Dc9sDfFqkpdMOz0hse1YscTux4TDPMQbDGQcY2w3waZGTTm9mACpObABpAfqCcQMDY5sEQS0Jfw4nzr95sOHgzwYGe4JaQA5jSGA7nLjhBmPDAd4GhkSCWiRnpyUzJLYdTtx4BuwXieQZhwlokbidZsz4A+iwecePP/74o8bGtr+9/RleLRhGMDAwk6J+FIyCUTAKRgFWAAA6CVAXH7Kl/QAAAABJRU5ErkJggg==","orcid":"","institution":"Indian Institute of Science","correspondingAuthor":true,"prefix":"","firstName":"Krishna","middleName":"Kant","lastName":"Mishra","suffix":""},{"id":483246504,"identity":"5bc00200-e10e-4617-82ec-2bdb5360ce1d","order_by":1,"name":"Rajnish Mallick","email":"","orcid":"","institution":"Indian Institute of Information Technology, Design and Manufacturing (IIITDM), Kancheepuram","correspondingAuthor":false,"prefix":"","firstName":"Rajnish","middleName":"","lastName":"Mallick","suffix":""},{"id":483246505,"identity":"0a9c471b-221c-4062-bd6b-e56b844ef77c","order_by":2,"name":"DineshKumar Harursampath","email":"","orcid":"","institution":"Indian Institute of Science","correspondingAuthor":false,"prefix":"","firstName":"DineshKumar","middleName":"","lastName":"Harursampath","suffix":""}],"badges":[],"createdAt":"2025-07-10 03:25:43","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-7088470/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7088470/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":86462800,"identity":"9c7bc3ff-9d0b-43b4-846f-33f7db63cf9e","added_by":"auto","created_at":"2025-07-11 02:25:14","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":711067,"visible":true,"origin":"","legend":"","description":"","filename":"researchsquare.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7088470/v1_covered_aaf914c2-1afc-4856-83cb-e06ccaf43705.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eA Mesh-Free Physics-Informed Neural Network Based on Rayleigh Quotient for Structural Frequency Analysis of Beams and Plates\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Indian Institute of Science Bangalore","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 (PINNs), Rayleigh Quotient, Structural Frequency Estimation, Euler-Bernoulli Beams, Kirchhoff-Love Plates, Deep Learning in Mechanics","lastPublishedDoi":"10.21203/rs.3.rs-7088470/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7088470/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAccurate estimation of natural frequencies is essential in structural engineering applications involving beams and plates. 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