DR-PDEE for engineered high-dimensional nonlinear stochastic systems: A physically-driven equation providing theoretical basis for data-driven approaches

DR-PDEE for engineered high-dimensional nonlinear stochastic systems: A physically-driven equation providing theoretical basis for data-driven approaches | 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 DR-PDEE for engineered high-dimensional nonlinear stochastic systems: A physically-driven equation providing theoretical basis for data-driven approaches Jian-Bing Chen, Ting-Ting Sun, Meng-Ze Lyu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4660971/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 06 Dec, 2024 Read the published version in Nonlinear Dynamics → Version 1 posted 9 You are reading this latest preprint version Abstract For over half a century, the analysis, control, and optimization design of high-dimensional nonlinear stochastic dynamical systems have posed long-standing challenges in the fields of science and engineering. Emerging scientific ideas and powerful technologies, such as big data and artificial intelligence (AI), offer new opportunity for addressing this problem. Data-driven techniques and AI methods are beginning to empower the research on stochastic dynamics. However, what is the physical essence, theoretical foundation, and effective applicable spectrum of data-driven and AI-aided (DDAA) stochastic dynamics? Answering this question has become important and urgent for advancing research in stochastic dynamics more solidly and effectively. This paper will provide a perspective on answering this question from the viewpoint of system dimensionality reduction. In the DDAA framework, the dimension of observed data of the studied system, such as the dimension of the complete state variables of the system, is fundamentally unknown. Thus, it can be considered that the stochastic dynamical systems under the DDAA framework are dimension-reduced subsystems of real-world systems. Therefore, a question of interest is: To what extent can the probability information predicted by the dimension-reduced subsystem characterize the probability information of the real-world system and serve as a decision basis? The paper will discuss issues such as the dimension-reduced probability density evolution equation (DR-PDEE) satisfied by the probability density function (PDF) of path-continuous non-Markov responses in general high-dimensional systems, the dimension-reduced partial integro-differential equation satisfied by the PDF of path-discontinuous responses, and the non-exchangeability of dimension reduction and imposition of absorbing boundary conditions. These studies suggest that the DR-PDEE and the dimension-reduced partial integro-differential equation can serve as important theoretical bases for the effectiveness and applicability boundaries of the DDAA framework. Data-driven and artificial-intelligence-aided (DDAA) Stochastic dynamics High-dimensional nonlinear stochastic dynamical system Dimension-reduced probability density evolution equation (DR-PDEE) Probability density function (PDF) Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 06 Dec, 2024 Read the published version in Nonlinear Dynamics → Version 1 posted Editorial decision: Revision requested 05 Sep, 2024 Reviews received at journal 05 Sep, 2024 Reviews received at journal 19 Aug, 2024 Reviewers agreed at journal 29 Jul, 2024 Reviewers agreed at journal 05 Jul, 2024 Reviewers invited by journal 03 Jul, 2024 Editor assigned by journal 02 Jul, 2024 Submission checks completed at journal 02 Jul, 2024 First submitted to journal 29 Jun, 2024 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-4660971","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":331425273,"identity":"0c983a28-c2b5-4801-9835-e9fa5e7fef72","order_by":0,"name":"Jian-Bing Chen","email":"","orcid":"","institution":"Tongji University","correspondingAuthor":false,"prefix":"","firstName":"Jian-Bing","middleName":"","lastName":"Chen","suffix":""},{"id":331425274,"identity":"f63ce2d4-a280-472c-a0e6-f0b377bedbfa","order_by":1,"name":"Ting-Ting Sun","email":"","orcid":"","institution":"Tongji University","correspondingAuthor":false,"prefix":"","firstName":"Ting-Ting","middleName":"","lastName":"Sun","suffix":""},{"id":331425275,"identity":"5bfe673a-cda5-41dc-bcb0-9274235ab095","order_by":2,"name":"Meng-Ze Lyu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAzUlEQVRIiWNgGAWjYBACPjBZASYTDjAwMBPWwgYmzxiQqoWxzQDGJ0aLRI7pxp/z/siZ8y94eIChwjqxgf3sAUJazG5IbjMwtpzxAOiwM+mJDTx5CYS1GG4zSNxw40DCAca2w4kNEjwGhLUkzoFp+UesloMNQC3nG4BaGojRwvOs7GbDMWNjgxvAQE44lm7cxpODXws/e/K2mz9q5OQMzp9J/vChxlq2n/0Mfi0MAglQhkROAgOIzYZfPciaAzDG8QN4lI2CUTAKRsFIBgBlJEm6wLzN2gAAAABJRU5ErkJggg==","orcid":"","institution":"Tongji University","correspondingAuthor":true,"prefix":"","firstName":"Meng-Ze","middleName":"","lastName":"Lyu","suffix":""}],"badges":[],"createdAt":"2024-06-30 02:23:26","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4660971/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4660971/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11071-024-10664-1","type":"published","date":"2024-12-06T15:58:12+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":70964908,"identity":"6201e284-e5ef-4365-b2d0-f1fcfd7b4261","added_by":"auto","created_at":"2024-12-09 16:17:20","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1780471,"visible":true,"origin":"","legend":"","description":"","filename":"240630NDSIsubmitted.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4660971/v1_covered_9f18ddd4-73dd-4aa7-8b9e-43b5f0827dde.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"DR-PDEE for engineered high-dimensional nonlinear stochastic systems: A physically-driven equation providing theoretical basis for data-driven approaches","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":"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":"Data-driven and artificial-intelligence-aided (DDAA), Stochastic dynamics, High-dimensional nonlinear stochastic dynamical system, Dimension-reduced probability density evolution equation (DR-PDEE), Probability density function (PDF)","lastPublishedDoi":"10.21203/rs.3.rs-4660971/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4660971/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eFor over half a century, the analysis, control, and optimization design of high-dimensional nonlinear stochastic dynamical systems have posed long-standing challenges in the fields of science and engineering. 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