Bi-level Identification of Governing Equations for Nonlinear Physical Systems

Bi-level Identification of Governing Equations for Nonlinear Physical Systems | 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 Article Bi-level Identification of Governing Equations for Nonlinear Physical Systems Lijun Yang, Zeyu Li, Huining Yuan, Wang Han, Hongjue Li, Yiming Hou, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4252008/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 09 May, 2025 Read the published version in Nature Computational Science → Version 1 posted You are reading this latest preprint version Abstract The identification of governing equations from data for nonlinear physical systems has been a long-standing challenge in scientific discovery. However, the inherently ill-posed nature of this inverse problem often leads to false discoveries that overfit certain datasets without capturing the true dynamics of the studied system. To overcome the challenge, we propose a Bi-level Identification of Equations (BILLIE) framework in this work to simultaneously perform equation discovery and validation in two hierarchical objectives of a bi-level optimization. BILLIE is effectively solved using policy gradient techniques from reinforcement learning and hence offers the possibility of identifying very challenging physical systems where the existing approaches fail. Our experimental results on the Navier-Stokes equation, the Burgers' equation, and the three-body system demonstrate BILLIE's dominant performance over existing methods. Moreover, we apply BILLIE to the task of discovering the RNA and protein velocity equations from real-world single-cell sequencing. For the first time, we can quantitatively discover these prominent velocity equations in a data-driven manner, which marks a departure from previous reliance on experts' empirical intuitions to hypothesize these equations. Notably, the equations identified by BILLIE surpass empirical equations in accurately characterizing the future states of differentiated cells. This demonstrates BILLIE's strong potential to contribute to the discovery of fundamental physical rules underlying diverse scientific phenomena. Physical sciences/Physics/Fluid dynamics Physical sciences/Mathematics and computing/Applied mathematics Physical sciences/Physics/Statistical physics, thermodynamics and nonlinear dynamics/Nonlinear phenomena Physical systems Equation identification Bi-level optimization Reinforcement learning Full Text Additional Declarations There is NO Competing Interest. Supplementary Files BILLIESI.pdf Cite Share Download PDF Status: Published Journal Publication published 09 May, 2025 Read the published version in Nature Computational Science → 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. 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-4252008","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":295814259,"identity":"fe0a4e6f-32f9-46f8-90e3-b2cbeba2ee9a","order_by":0,"name":"Lijun 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