Data-Driven Koopman Operator Approximations for Hysteresis in Piezoelectric Composites

Data-Driven Koopman Operator Approximations for Hysteresis in Piezoelectric Composites | 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 Data-Driven Koopman Operator Approximations for Hysteresis in Piezoelectric Composites Abdulaziz Alazemi, Andrew Kurdila This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9260796/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract This paper addresses the challenges associated with modeling and predicting the behavior of hysteretically nonlinear piezoelectric composite beams, which exhibit complex, high-dimensional dynamics and involve parameters that are often poorly characterized. Traditional modeling approaches can struggle to capture the memory-dependent nonlinearities inherent in such systems.To address these challenges, we derive novel error bounds for data-driven approximations of Koopman operators in discrete time. The proposed approach employs a vector-valued reproducing kernel Hilbert space (vRKHS) induced by a general, potentially non-diagonal operator-valued kernel, thereby extending existing Koopman approximation methods based on scalar-valued kernels. Using tools from the theory of inverse problems, the resulting error bounds explicitly decompose the total approximation error into a sample error contribution, which depends on noise in the observations, and an approximation error contribution, which depends on the dimension of the approximating subspace. The bounds reveal a fundamental trade-off in which approximation error decreases while sample error may increase as the approximation space is enriched.The theoretical results are investigated through a numerical case study involving a hysteretically nonlinear piezoelectric composite beam modeled as a functional differential equation (FDE). The numerical results demonstrate that the relative $L^\infty(\Omega)$ prediction error exhibits the qualitative behavior predicted by the theory, with approximation error dominating for coarse approximations and sample error dominating for highly refined approximations in the presence of noise. These results illustrate how data-driven Koopman approximations can capture the dominant dynamics of complex hysteretic systems using low-dimensional discrete-time models. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 30 Mar, 2026 Editor assigned by journal 30 Mar, 2026 Submission checks completed at journal 30 Mar, 2026 First submitted to journal 29 Mar, 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. 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-9260796","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":614734878,"identity":"3b0d22b4-3746-4579-ab84-27bf0c33425b","order_by":0,"name":"Abdulaziz Alazemi","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVRIiWNgGAWjYBACNnYGAyAlkQAkGB+AhQ4Q0MLPjNDCbMCQQIQWyWawFrBaNgmitBgcZt4mzVNjkcfP3v6smvcHgxzfjQRCWtjKpHmOSRRL9pwxu82TwGAsSVgLj5l0DptE4oYbOWw3ZyQwABkEtNiDtfyTSNx///mzQqCWeoJawLbktgFtkWAwY/iQwJBgQIRfiq3/9kkkzjiTYyzxIU3CcOaZBwS0HG/eeHPGt7rE/vbjDz8k2NjI8x0nYAs6kCBN+SgYBaNgFIwC7AAAhmxELmy4AH4AAAAASUVORK5CYII=","orcid":"","institution":"Virginia Tech","correspondingAuthor":true,"prefix":"","firstName":"Abdulaziz","middleName":"","lastName":"Alazemi","suffix":""},{"id":614734881,"identity":"2dd95168-5cca-4216-8520-ecb68502174e","order_by":1,"name":"Andrew Kurdila","email":"","orcid":"","institution":"Virginia Tech","correspondingAuthor":false,"prefix":"","firstName":"Andrew","middleName":"","lastName":"Kurdila","suffix":""}],"badges":[],"createdAt":"2026-03-29 20:08:15","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9260796/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9260796/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107486385,"identity":"bd89fccd-5af0-48b3-9cad-12e244d399ff","added_by":"auto","created_at":"2026-04-22 02:38:13","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1163371,"visible":true,"origin":"","legend":"","description":"","filename":"DataDrivenKoopmanOperatorApproximationsforHysteresisinPiezoelectricComposites4.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9260796/v1_covered_57150f12-2d1f-482b-ba34-fa1c2bc25bea.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Data-Driven Koopman Operator Approximations for Hysteresis in Piezoelectric Composites","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":"nonlinear-dynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"nody","sideBox":"Learn more about [Nonlinear Dynamics](https://www.springer.com/journal/11071)","snPcode":"11071","submissionUrl":"https://submission.nature.com/new-submission/11071/3","title":"Nonlinear Dynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-9260796/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9260796/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This paper addresses the challenges associated with modeling and predicting the behavior of hysteretically nonlinear piezoelectric composite beams, which exhibit complex, high-dimensional dynamics and involve parameters that are often poorly characterized. Traditional modeling approaches can struggle to capture the memory-dependent nonlinearities inherent in such systems.To address these challenges, we derive novel error bounds for data-driven approximations of Koopman operators in discrete time. The proposed approach employs a vector-valued reproducing kernel Hilbert space (vRKHS) induced by a general, potentially non-diagonal operator-valued kernel, thereby extending existing Koopman approximation methods based on scalar-valued kernels. Using tools from the theory of inverse problems, the resulting error bounds explicitly decompose the total approximation error into a sample error contribution, which depends on noise in the observations, and an approximation error contribution, which depends on the dimension of the approximating subspace. The bounds reveal a fundamental trade-off in which approximation error decreases while sample error may increase as the approximation space is enriched.The theoretical results are investigated through a numerical case study involving a hysteretically nonlinear piezoelectric composite beam modeled as a functional differential equation (FDE). The numerical results demonstrate that the relative $L^\\infty(\\Omega)$ prediction error exhibits the qualitative behavior predicted by the theory, with approximation error dominating for coarse approximations and sample error dominating for highly refined approximations in the presence of noise. 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