Data-driven solutions of the nonlinear Boussinesq and Schrödinger equations of second order in time with physics-informed neural networks

Data-driven solutions of the nonlinear Boussinesq and Schrödinger equations of second order in time with physics-informed neural networks | 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 solutions of the nonlinear Boussinesq and Schrödinger equations of second order in time with physics-informed neural networks Wei Hu, Chao Dong, Shaolong Zheng, Ruozhou Gao, Hongyu Wu, Hongbo Zhang, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3974638/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Deep learning combining the physics information is employed to solve the Boussinesq and Schrödinger equations with second-order time derivative. High prediction accuracies are achieved by adding a new initial loss term in the physics-informed neural networks along with the adaptive activation function and loss-balanced coefficients. The numerical simulations are carried out with different initial and boundary conditions, in which the relative L 2 -norm errors are all around 10 -4 . The prediction accuracies have been improved by two orders of magnitude compared to the former results in certain simulations. The dynamic behavior of solitons and their interaction are studied in the colliding and chasing process for the Boussinesq equation. More training time is needed for the solver of the Boussinesq equation when the width of the two-soliton solutions is narrower with other parameters fixed. Simulations in large domains are conducted to explore soliton and rogue wave solutions of the Schrödinger equation and the high precisions are achieved. Improved physics-informed neural networks Boussinesq equation Schrödinger equation Soliton wave Rogue wave Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted 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-3974638","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":274969920,"identity":"a284587e-0b54-40d6-8a4d-799e5476a802","order_by":0,"name":"Wei Hu","email":"","orcid":"","institution":"Lishui University","correspondingAuthor":false,"prefix":"","firstName":"Wei","middleName":"","lastName":"Hu","suffix":""},{"id":274969921,"identity":"650a89a1-2fb7-49a0-9908-20fb33ab6164","order_by":1,"name":"Chao 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