Solving Fokker-Planck equations using deep KD-tree with a small amount of data

Solving Fokker-Planck equations using deep KD-tree with a small amount of data | 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 Solving Fokker-Planck equations using deep KD-tree with a small amount of data Yong Xu, Hao Zhang, Qi Liu, Xiaolong Wang, Yongge Li This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-1265549/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 5 You are reading this latest preprint version Abstract The Fokker-Planck (FP) equation can deterministically describe the evolution of the probability density function, which plays an extremely significant role in the fields of stochastic dynamics. Unfortunately, the limited samples that arise from the consideration of engineering practice are inevitable, which restricts the solving of the FP equation. Accordingly, in the present study, a super-DL-FP framework is established to solve the steady-state FP equation with a small amount of data, through combining the deep KD-tree and the DLFP approach proposed in [Chaos 30, 013133 (2020)]. It should be emphasized that the normalization condition is of great importance and have to be considered in solving the steady-state FP equation. An appropriate integral estimation for the normalization condition under non-uniform meshing can effectively improve the precision of the solution, but it is still a challenging problem, especially for the case of small data. Thus, the so-called deep KD-tree method is innovatively proposed to estimate the normalized integral with a small random dataset. The main target is to obtain the appropriate discrete integral points and corresponding integral volumes by executing multiple KD-tree segmentation based on random data on the integral region. Several numerical experiments and comparisons are implemented to illustrate the superior performance of the super-DL-FP method. The obtained results indicate that the proposed algorithm can accomplish higher accuracy in the sense of lower cost than the well-known algorithms like center difference scheme, Chebyshev spectrum algorithm, and normalized flow approach. Fokker-Planck equation deep KD-tree deep learning super-DL-FP Full Text Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 30 Jan, 2022 Reviewers invited by journal 30 Jan, 2022 Editor assigned by journal 29 Jan, 2022 First submitted to journal 28 Jan, 2022 Editorial decision: Major revisions 20 Jan, 2022 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-1265549","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":80212567,"identity":"b469b4c8-7ae9-4a74-9780-ce4a5f6743ef","order_by":0,"name":"Yong Xu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwElEQVRIiWNgGAWjYBACAyBm/FAB5fEQq4VZ4gypWhh420jRYs5/xkxCct6dxO0SCYwP3rYxyJsT0mI5I8fYoHDbs8SdMxKYDee2MRjubCDksBs8hg8ktx3O3XAjgU0a6MIEgwOEtJw/Y3CAdw5YC/tv4rQcyDF8wNsAsYWZOC030oqNJY4drt/Z87BZcs45CcMNhB12eJvkh5rDxubsyQc/vCmzkSdoC0IvA2MDkJIgVj0DNE5HwSgYBaNgFGADANA2QbgvmcisAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0002-8407-4650","institution":"Northwestern Polytechnical University","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Yong","middleName":"","lastName":"Xu","suffix":""},{"id":80212566,"identity":"df525ce6-cc33-4b0f-95cb-747c837a7374","order_by":1,"name":"Hao Zhang","email":"","orcid":"","institution":"Northwestern Polytechnical University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Hao","middleName":"","lastName":"Zhang","suffix":""},{"id":80212568,"identity":"6239b0b1-130c-4231-8b62-851e1c2d7d14","order_by":2,"name":"Qi Liu","email":"","orcid":"","institution":"Northwestern Polytechnical University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Qi","middleName":"","lastName":"Liu","suffix":""},{"id":80212569,"identity":"634977b3-0305-48b4-8371-22d224231e24","order_by":3,"name":"Xiaolong Wang","email":"","orcid":"","institution":"Shaanxi Normal University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Xiaolong","middleName":"","lastName":"Wang","suffix":""},{"id":80212570,"identity":"1d7c18c0-f448-4f7d-a9b5-21fd030e91e2","order_by":4,"name":"Yongge Li","email":"","orcid":"","institution":"Northwestern Polytechnical University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Yongge","middleName":"","lastName":"Li","suffix":""}],"badges":[],"createdAt":"2022-01-16 11:27:06","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-1265549/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-1265549/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":17808637,"identity":"ebd0958e-8aa6-4ea9-bf6e-4b0c99491454","added_by":"auto","created_at":"2022-01-31 15:41:38","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1810956,"visible":true,"origin":"","legend":"","description":"","filename":"template.pdf","url":"https://assets-eu.researchsquare.com/files/rs-1265549/v1_covered.pdf"}],"financialInterests":"","formattedTitle":"Solving Fokker-Planck equations using deep KD-tree with a small amount of data","fulltext":[{"header":"Full Text","content":"This preprint is available for \u003ca href='/article/rs-1265549/latest.pdf' target='_blank'\u003edownload as a PDF\u003c/a\u003e."}],"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":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nonlinear-dynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"nody","sideBox":"Learn more about [Nonlinear Dynamics](https://www.springer.com/journal/11071)","snPcode":"11071","submissionUrl":"https://submission.nature.com/new-submission/11071/3","title":"Nonlinear Dynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Fokker-Planck equation , deep KD-tree , deep learning , super-DL-FP","lastPublishedDoi":"10.21203/rs.3.rs-1265549/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-1265549/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"The Fokker-Planck (FP) equation can deterministically describe the evolution of the probability density function, which plays an extremely significant role in the fields of stochastic dynamics. 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The main target is to obtain the appropriate discrete integral points and corresponding integral volumes by executing multiple KD-tree segmentation based on random data on the integral region. Several numerical experiments and comparisons are implemented to illustrate the superior performance of the super-DL-FP method. 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