Physics-informed Neural Network with Hard Constraints based on Dual-Step Training Strategy for Periodic Orbits in the CR3BP

Physics-informed Neural Network with Hard Constraints based on Dual-Step Training Strategy for Periodic Orbits in the CR3BP | 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 Physics-informed Neural Network with Hard Constraints based on Dual-Step Training Strategy for Periodic Orbits in the CR3BP Yong Chen, Ming Cui, Ying-Jing Qian, Ye-Feng Cheng, Wen-Xue Chen This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8772924/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 study focuses on determining periodic orbits in the Circular Restricted Three-Body Problem (CR3BP) by introducing a novel Physics-informed Neural Network framework with hard constraints based on Dual-Step Training (HDT-PINN). The proposed approach integrates several key components: initial hard constraints, a dual-step training strategy, and an augmented parameter optimization. The initial hard constraints enforce consistency between the computed orbit and the target orbit for the initial conditions. The dual-step training strategy, including the pre-training step and final training step, effectively avoids convergence to local minima and is crucial for the successful identification of target periodic orbits. Furthermore, the augmented parameter optimization incorporates the initial velocity $\bar{\dot{y}}_0$ and the orbital period $t_f$ as trainable parameters, significantly enhancing the accuracy of the solutions. The effectiveness and robustness of the method are demonstrated across four distinct families of periodic orbits: Lyapunov orbits around the $L_1$ and $L_2$ points, distant retrograde orbits (DROs), and halo orbits. Furthermore, we conducted a convergence analysis of the initial velocity $\bar{\dot{y} }_0$ under various levels of disturbance and determined the convergence region for the initial velocity error $\delta_1$ associated with the periodic orbit considered in the example. The method proposed in this study exhibits a significantly broader convergence region in initial velocity error and precise target periodic orbit. Physics-informed neural network Dual-step training strategy CR3BP Periodic orbit Parameter optimization Full Text Additional Declarations No competing interests reported. Supplementary Files nonlieardynamic.zip Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 04 Feb, 2026 Editor assigned by journal 03 Feb, 2026 Submission checks completed at journal 03 Feb, 2026 First submitted to journal 03 Feb, 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-8772924","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":585510621,"identity":"6444e8eb-d594-4e72-8ef7-55ef4fa81f65","order_by":0,"name":"Yong Chen","email":"","orcid":"","institution":"Beijing University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Yong","middleName":"","lastName":"Chen","suffix":""},{"id":585510623,"identity":"44ddc174-94fe-46cb-a790-018423ffed4b","order_by":1,"name":"Ming Cui","email":"","orcid":"","institution":"Beijing University of 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