Neural Networks for Structured Grid Generation

Neural Networks for Structured Grid Generation | 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 Neural Networks for Structured Grid Generation Bari Khairullin, Sergey Rykovanov, Rishat Zagidullin This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5819666/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 Numerical solutions of partial differential equations (PDEs) on regular domains provide simplicity as we can rely on the structure of the space. We investigate a novel neural network (NN) - based approach to generate 2-dimensional body-fitted curvilinear coordinate systems (BFCs) that allow to stay on regular grids even when the complex geometry is considered. We describe a feed-forward neural network (FNN) as a geometric transformation that can represent a diffeomorphism under certain constraints and approximations, followed by the ways of training it to create BFCs. We show that the optimization system is similar to a physics informed neural network (PINN) based solution of Winslow equations. Unlike in classical BFC generation, FNN provides a differentiable mapping between spaces, and all the Jacobian matrices may be obtained exactly at any given point. Also, it allows to change an interior nodes distribution without the need of recreating the whole mapping. 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-5819666","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":402651794,"identity":"b63c64b5-e2f4-4632-8c6b-e1660976d769","order_by":0,"name":"Bari Khairullin","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCklEQVRIie2QsWrDMBCGLxg8ieQBCvErSGRu8ioyhnhpSiBLhg6e1MWQNVtewX2DCwJPMlk9ZHAXz3qAEnIyhC5ys3bQBwLp5z6dTgCBwD9kRQsBEWKACXYc5sBoQ6kcU0TxqwBKDgtS3Hlc4e7CoWRQANLimSI+m28Ec02mL4aj3F7zU6lR2/3yHSjxzsJyatT2Qk3fSOH9pmqURDTZDijxz7KmWayeKMacojfVjDZnhfRC5lX4pR+U1UPJk4NTbn8orevS6vShSGhKUopxRRypizQ6UyzeOkVUppZo6ix1iXf82TqyttavpzL66uyPTpIyO3f7j2V6YFHl/WWH9KbxaH0gEAgEnnEHKThz+VV7hRwAAAAASUVORK5CYII=","orcid":"","institution":"Skolkovo Institute of Science and Technology","correspondingAuthor":true,"prefix":"","firstName":"Bari","middleName":"","lastName":"Khairullin","suffix":""},{"id":402651795,"identity":"2cb7135a-2abf-419b-836d-7b53f1175ad4","order_by":1,"name":"Sergey Rykovanov","email":"","orcid":"","institution":"Skolkovo Institute of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Sergey","middleName":"","lastName":"Rykovanov","suffix":""},{"id":402651796,"identity":"e1700195-62eb-4dbc-8706-6eec52b258f3","order_by":2,"name":"Rishat Zagidullin","email":"","orcid":"","institution":"Skolkovo Institute of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Rishat","middleName":"","lastName":"Zagidullin","suffix":""}],"badges":[],"createdAt":"2025-01-13 11:53:23","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5819666/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5819666/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":74023237,"identity":"25cc488f-f9ec-43c4-8d2b-bf0a226da90d","added_by":"auto","created_at":"2025-01-17 05:34:57","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":953932,"visible":true,"origin":"","legend":"","description":"","filename":"toupload.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5819666/v1_covered_a86535f7-5534-4bcd-847b-5ac5ab360b66.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Neural Networks for Structured Grid Generation","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-5819666/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5819666/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Numerical solutions of partial differential equations (PDEs) on regular domains provide simplicity as we can rely on the structure of the space. 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