Solving the Richards Infiltration Equation by Coupling Physics-Informed Neural Networks with Hydrus-1D

Solving the Richards Infiltration Equation by Coupling Physics-Informed Neural Networks with Hydrus-1D | 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 Solving the Richards Infiltration Equation by Coupling Physics-Informed Neural Networks with Hydrus-1D Yanling Li, Qianxing Sun, Yuliang Fu, Junfang Wei This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6195280/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 28 May, 2025 Read the published version in Scientific Reports → Version 1 posted 20 You are reading this latest preprint version Abstract The movement and infiltration of groundwater play a crucial role in environmental engineering and water resource management. The Richards equation, a fundamental model describing water transport in unsaturated soils, encounters significant challenges in traditional numerical solutions due to its strong nonlinearity, complex boundary conditions, and computational inefficiency. To address these issues, this study proposes an improved physics-informed neural network (PINN) method based on data fusion. This approach is designed to handle the intricate boundary conditions and nonlinear water diffusion characteristics in groundwater seepage by integrating data with physical constraints, thereby forming a dual-driven solution framework that leverages both data and physics. The proposed improved algorithm integrates Hydrus data, leveraging a small portion of data to reduce the model's dependence on parameter initialization. Simultaneously, it enables the model to automatically adjust to variations in physical processes under different data conditions, thereby enhancing the accuracy and stability of the solution. Experimental results demonstrate the strong generalization ability of this method, particularly in data-scarce regions, where physical constraints ensure the reliability of the model's solutions. Earth and environmental sciences/Hydrology Physical sciences/Mathematics and computing/Applied mathematics Physical sciences/Mathematics and computing/Computer science Richards equation PINN algorithm Hydrus-1D dual-driven model Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 28 May, 2025 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 14 Apr, 2025 Reviews received at journal 14 Apr, 2025 Reviews received at journal 13 Apr, 2025 Reviews received at journal 13 Apr, 2025 Reviews received at journal 10 Apr, 2025 Reviews received at journal 09 Apr, 2025 Reviewers agreed at journal 07 Apr, 2025 Reviewers agreed at journal 07 Apr, 2025 Reviewers agreed at journal 07 Apr, 2025 Reviewers agreed at journal 06 Apr, 2025 Reviewers agreed at journal 06 Apr, 2025 Reviewers agreed at journal 06 Apr, 2025 Reviewers agreed at journal 05 Apr, 2025 Reviewers agreed at journal 04 Apr, 2025 Reviewers agreed at journal 04 Apr, 2025 Reviewers invited by journal 04 Apr, 2025 Editor assigned by journal 04 Apr, 2025 Editor invited by journal 26 Mar, 2025 Submission checks completed at journal 20 Mar, 2025 First submitted to journal 20 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. 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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-6195280","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":439133937,"identity":"df4a0fb6-5906-42f8-8553-d5199c67f315","order_by":0,"name":"Yanling Li","email":"","orcid":"","institution":"North China University of Water Resources and Electric","correspondingAuthor":false,"prefix":"","firstName":"Yanling","middleName":"","lastName":"Li","suffix":""},{"id":439133938,"identity":"2d4a82bc-30bd-4886-8d23-f1952ed844b9","order_by":1,"name":"Qianxing Sun","email":"","orcid":"","institution":"North China University of Water Resources and Electric","correspondingAuthor":false,"prefix":"","firstName":"Qianxing","middleName":"","lastName":"Sun","suffix":""},{"id":439133939,"identity":"8d73e3b6-d895-4953-bc0f-ee5c938344a2","order_by":2,"name":"Yuliang Fu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAxElEQVRIiWNgGAWjYBACxvbGxsc//7DJ8ROthbnncLMxYwOfsWQDsVrYZ6S3STM2yCVuOECsFt6GxAbpwh1mxsbHkzcw/KjYRliLZMPBBuOZZ9LkzM48K2DsOXObsBbDxsaGBB62Y8ZmN3IMmBnbiNBif5ix4QAP2//EzTOI1cLYxtjYzNvGlrhBgmgtPYzNjDPOsBlLAP1ykCi/MM5//vzHhwpgVLYnb3zwo4IILUggweAASerBWkjVMQpGwSgYBSMEAADPhUHhYfgKXgAAAABJRU5ErkJggg==","orcid":"","institution":"North China University of Water Resources and Electric","correspondingAuthor":true,"prefix":"","firstName":"Yuliang","middleName":"","lastName":"Fu","suffix":""},{"id":439133941,"identity":"b2bc631d-853e-467c-bfed-15b192df9e06","order_by":3,"name":"Junfang Wei","email":"","orcid":"","institution":"North China University of Water Resources and Electric","correspondingAuthor":false,"prefix":"","firstName":"Junfang","middleName":"","lastName":"Wei","suffix":""}],"badges":[],"createdAt":"2025-03-10 11:53:28","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6195280/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6195280/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41598-025-02978-w","type":"published","date":"2025-05-28T15:57:00+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":83782867,"identity":"9d5e2b89-4dc0-464a-b539-25dd9517b9b7","added_by":"auto","created_at":"2025-06-02 16:07:46","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1003508,"visible":true,"origin":"","legend":"","description":"","filename":"manuscipt2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6195280/v1_covered_2ebf23e0-5b2e-4c0e-9445-c3bb20026839.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Solving the Richards Infiltration Equation by Coupling Physics-Informed Neural Networks with Hydrus-1D","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Richards equation, PINN algorithm, Hydrus-1D, dual-driven model","lastPublishedDoi":"10.21203/rs.3.rs-6195280/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6195280/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe movement and infiltration of groundwater play a crucial role in environmental engineering and water resource management. 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