Discrete-Time Dynamical Systems Characterization via Invariance and Approximation of Koopman Operators and Operator-Valued Kernels

Discrete-Time Dynamical Systems Characterization via Invariance and Approximation of Koopman Operators and Operator-Valued Kernels | 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 Discrete-Time Dynamical Systems Characterization via Invariance and Approximation of Koopman Operators and Operator-Valued Kernels Soonyong Yang, Andrew Kurdila, Andrea L'Afflitto, Rushikesh Kamalapurkar, and 3 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7566917/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 24 Mar, 2026 Read the published version in Nonlinear Dynamics → Version 1 posted 12 You are reading this latest preprint version Abstract This paper presents approaches based on native space theory and Koopman operators to characterize the dynamics of nonlinear, discrete-time dynamical models employing measured data only. Given approximation schemes of the plant dynamics based on Koopman operators contained in vector-valued reproducing kernel Hilbert spaces (vRKHSs), we deduce rates of convergence for these schemes. In particular, we present a necessary and sufficient condition for Koopman invariance of observables in vRKHSs that are defined via generic non-diagonal operator-valued kernels, and develop sufficient conditions to guarantee the Koopman invariance for vRKHSs defined in terms of a class of diagonal operator-valued kernels. Principles of inverse problems are leveraged to derive error bounds for approximations of the Koopman operator that include both a deterministic sampling error and an approximation error term. The deterministic sampling error arises since imprecisely measured samples are used to approximate the Koopman operator. This work is the first to present overall bounds in the deterministic setting that explicitly account for the sampling error, which, in general, increases with the reduced dimension. Numerical examples illustrate the proposed results. Mathematics Subject Classification (2020) 46E22 · 47A58 · 47B32 Koopman theory inverse problems Koopman invariance approximation Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 24 Mar, 2026 Read the published version in Nonlinear Dynamics → Version 1 posted Editorial decision: Revision requested 11 Dec, 2025 Reviews received at journal 01 Dec, 2025 Reviews received at journal 27 Nov, 2025 Reviews received at journal 26 Nov, 2025 Reviewers agreed at journal 31 Oct, 2025 Reviewers agreed at journal 31 Oct, 2025 Reviewers agreed at journal 29 Oct, 2025 Reviewers agreed at journal 29 Oct, 2025 Reviewers invited by journal 13 Oct, 2025 Editor assigned by journal 20 Sep, 2025 Submission checks completed at journal 15 Sep, 2025 First submitted to journal 08 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. 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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-7566917","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":533921963,"identity":"841ffc22-0cc5-4947-bd94-4aa8a269f899","order_by":0,"name":"Soonyong Yang","email":"","orcid":"","institution":"Virginia Tech","correspondingAuthor":false,"prefix":"","firstName":"Soonyong","middleName":"","lastName":"Yang","suffix":""},{"id":533921964,"identity":"deb8145a-d9d7-4a68-bb38-712ac95e641b","order_by":1,"name":"Andrew Kurdila","email":"","orcid":"","institution":"Virginia 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Given approximation schemes of the plant dynamics based on Koopman operators contained in vector-valued reproducing kernel Hilbert spaces (vRKHSs), we deduce rates of convergence for these schemes. In particular, we present a necessary and sufficient condition for Koopman invariance of observables in vRKHSs that are defined via generic non-diagonal operator-valued kernels, and develop sufficient conditions to guarantee the Koopman invariance for vRKHSs defined in terms of a class of diagonal operator-valued kernels.\u0026nbsp; Principles of inverse problems are leveraged to derive error bounds for approximations of the Koopman operator that include both a deterministic sampling error and an approximation error term. 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