Sparse SGNN for Interpretable Learning of Stationary Solutions of Fokker-Planck-Kolmogorov Equations

Sparse SGNN for Interpretable Learning of Stationary Solutions of Fokker-Planck-Kolmogorov Equations | 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 Sparse SGNN for Interpretable Learning of Stationary Solutions of Fokker-Planck-Kolmogorov Equations Wen Yu, Yeyin Xu, Xi Wang, Ling Hong, Jun Jiang, Siyuan Xing This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6212909/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 12 Aug, 2025 Read the published version in Nonlinear Dynamics → Version 1 posted 4 You are reading this latest preprint version Abstract This paper introduces an interpretable machine learning framework that combines Separable Gaussian Neural Networks (SGNN) and sparse regression to solve stationary solutions of FPK equations. The structure of SGNN naturally satisfies the boundary conditions of FPK equations at infinity, which MLP-based Physics-informed Neural networks (PINNs) can only approximate. In addition, it enables the transformation of probability normalization condition into a weight constraint that can be encoded into the loss function. This eliminates the need for computationally expensive sampling, therefore, significantly reducing computational overhead. The integration with sparse regression into training process yields parsimonious SGNN models that provide highly interpretable predictions in the form of Gaussian radial-basis functions. We demonstrate our framework's effectiveness on several challenging examples including the Duffing oscillator, Van der Pol oscillator, and Lorenz system, achieving approximately 60%-98% reduction in network size while not only maintaining but improving prediction accuracy. Parametric study reveals the optimal balance between model sparsity and accuracy through studies of pruning threshold and sparsity-promoting parameter. At last, we show the limitation of the proposed approach for systems with multi-modal-annular distributions and propose a masking-based post-training scheme to address this limitation. This work demonstrates that combining SGNN with sparse optimization techniques can significantly advance our ability to solve FPK equations efficiently while maintaining physical interpretability. SGNN Sparse regression FPK equation Stationary solutions Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 12 Aug, 2025 Read the published version in Nonlinear Dynamics → Version 1 posted Reviewers invited by journal 18 Mar, 2025 Editor assigned by journal 13 Mar, 2025 Submission checks completed at journal 13 Mar, 2025 First submitted to journal 12 Mar, 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-6212909","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":431851074,"identity":"384fb8a8-7c72-4bca-9eda-4e183782db78","order_by":0,"name":"Wen Yu","email":"","orcid":"","institution":"Xi’an Jiaotong University","correspondingAuthor":false,"prefix":"","firstName":"Wen","middleName":"","lastName":"Yu","suffix":""},{"id":431851075,"identity":"e39f17bf-c4bd-45a7-82fc-9274c2079323","order_by":1,"name":"Yeyin Xu","email":"","orcid":"","institution":"Xi’an Jiaotong 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