A new solution method for high-dimensionalstochastic dynamical systems via delay embedding

A new solution method for high-dimensionalstochastic dynamical systems via delay embedding | 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 A new solution method for high-dimensionalstochastic dynamical systems via delay embedding Xinyi Li, Liang Wang, Zhonghua Zhang, Minjuan Yuan, Jiahui Peng This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8668363/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 Stochastic phenomena are pervasive in natural and engineering systems, andthe solution of high-dimensional stochastic dynamics remains a fundamentalchallenge. In many cases, only a subset of system variables is of primary interest. Hence, dimensionality reduction methods provide a viable means to studysuch high-dimensional systems. In this study, we propose a data-driven computational framework for high-dimensional stochastic dynamical systems basedon the concept of delay embedding. The method constructs a family of delayembedding mappings from multiple time series of the original system and incorporates them into the governing equations, thereby approximately transformingthe high-dimensional dynamics into a low-dimensional time-delay system. Theeffectiveness of the proposed framework is demonstrated through probability density function analysis of two four-dimensional systems and one ten-dimensionalsystem subjected to Gaussian white noise excitation. Numerical results show thatthe method achieves accurate agreement with Monte Carlo simulations while substantially reducing computational cost, offering a powerful tool for the efficientanalysis of high-dimensional stochastic systems. High-dimensional stochastic systems Probability density function Delay embedding 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. 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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-8668363","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":586528016,"identity":"7c35edbd-58dc-4e8d-b703-a411952d4c13","order_by":0,"name":"Xinyi Li","email":"","orcid":"","institution":"Northwestern Polytechnical University","correspondingAuthor":false,"prefix":"","firstName":"Xinyi","middleName":"","lastName":"Li","suffix":""},{"id":586528017,"identity":"940771c6-436d-4298-9d27-08c752b8f6e4","order_by":1,"name":"Liang Wang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABD0lEQVRIiWNgGAWjYBACPmYYi5mx8cGHCiiHB48WNrgWduZmwxlniNECZ/Gzt0nzthGjhZ352cOvbXZ58s6MDdK88+4kbpdIYHzwto1B3hynw9jMjWXbkosNDzM2GM7d9ixx54wEZsO5bQyGOxtw+sVMWnIbc+LGZsaGhLfbDuduuJHABnJhgsEBXFrYvwG11IO1HOCdA9bC/hu/Fh4zyY/bDifOBwZyI28DxBZmAlrKpBn/HU/cwMzYzDjj2OH6nT0PmyXnnJMw3IBDCz//8W2SP85UJ87vP/78x4eaw8bm7MkHP7wps5HHZQsIMINiAa7AgIGxAUhJ4FYPBIw/gIR8A1zLKBgFo2AUjAJUAAB9f1wz2wJchQAAAABJRU5ErkJggg==","orcid":"","institution":"Northwestern Polytechnical University","correspondingAuthor":true,"prefix":"","firstName":"Liang","middleName":"","lastName":"Wang","suffix":""},{"id":586528018,"identity":"39b27f0f-310f-42d7-8d5e-074ccee314d2","order_by":2,"name":"Zhonghua Zhang","email":"","orcid":"","institution":"Northwestern Polytechnical University","correspondingAuthor":false,"prefix":"","firstName":"Zhonghua","middleName":"","lastName":"Zhang","suffix":""},{"id":586528019,"identity":"7dc1692c-60e4-4262-8b0f-246ddb863763","order_by":3,"name":"Minjuan Yuan","email":"","orcid":"","institution":"Xi’an University of Posts and Telecommunications","correspondingAuthor":false,"prefix":"","firstName":"Minjuan","middleName":"","lastName":"Yuan","suffix":""},{"id":586528020,"identity":"1fb5d915-c026-48f0-9672-17d9e21d282f","order_by":4,"name":"Jiahui Peng","email":"","orcid":"","institution":"Xinjiang University","correspondingAuthor":false,"prefix":"","firstName":"Jiahui","middleName":"","lastName":"Peng","suffix":""}],"badges":[],"createdAt":"2026-01-22 10:23:15","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8668363/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8668363/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104400864,"identity":"e1affb64-272e-4c7b-ab70-a3d0ed344cb4","added_by":"auto","created_at":"2026-03-11 12:11:16","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":17029853,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8668363/v1_covered_e6ab99d2-6c3b-4577-8b0f-c081da2542fb.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A new solution method for high-dimensionalstochastic dynamical systems via delay embedding","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"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":"High-dimensional stochastic systems, Probability density function, Delay embedding","lastPublishedDoi":"10.21203/rs.3.rs-8668363/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8668363/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eStochastic phenomena are pervasive in natural and engineering systems, andthe solution of high-dimensional stochastic dynamics remains a fundamentalchallenge. 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