Universal differential equations for optimal control problems and its application on cancer therapy

Universal differential equations for optimal control problems and its application on cancer therapy | 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 Universal differential equations for optimal control problems and its application on cancer therapy Wenjing Zhang, Wandi Ding, Huaiping Zhu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6855591/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 This paper highlights a parallel between the forward–backward sweeping method for optimal control and deep learning training procedures. We reformulate a classical optimal control problem, constrained by a differential equation system, into an optimization framework that uses neural networks to represent control variables. We demonstrate that this deep learning method adheres to Pontryagin’s Maximum Principle and mitigates numerical instabilities by employing backward propagation instead of a backward sweep for the adjoint equations. As a case study, we solve an optimal control problem to find the optimal combination of immunotherapy and chemotherapy. Our approach holds significant potential across various fields, including epidemiology, ecological modeling, engineering, and financial mathematics, where optimal control under complex dynamic constraints is crucial. Optimal control scientific machine learning universal differential equations cancer therapy 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-6855591","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":495853936,"identity":"28a07316-c055-40b6-8e2e-a73f7df762c3","order_by":0,"name":"Wenjing Zhang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7klEQVRIiWNgGAWjYJACgwQQKQEiKiAiEiRoOWMgQZQWhMmMbURoMTjee6Dg4Y5aBvnZzc8efp33p87gAPPB2zz4tJw5l2CQeOY4A+OcY+bGstsMJAwOsCVb49NidiPHwCCx7RgDs0SCmbQkWAuPmTReLfffQLSwSaR/k5acA9LC/w2/lhs8IC01DDwSOWaSHxvAtrDh1WJ/BuywAzwSEjll0gzHjCVnHmYztpyDR4tk+xkzw59tdXLyM9K3Sf6okePnO9788MYbPFqAgM2AgeEw2CXMEBK/crCSBwwMdWAW4w/CqkfBKBgFo2AEAgDfd0eTxEIy4wAAAABJRU5ErkJggg==","orcid":"","institution":"Texas Tech University","correspondingAuthor":true,"prefix":"","firstName":"Wenjing","middleName":"","lastName":"Zhang","suffix":""},{"id":495853937,"identity":"b040ac9f-2c07-40bf-ae05-670f4b25cc51","order_by":1,"name":"Wandi Ding","email":"","orcid":"","institution":"Middle Tennessee State University","correspondingAuthor":false,"prefix":"","firstName":"Wandi","middleName":"","lastName":"Ding","suffix":""},{"id":495853938,"identity":"ded0b115-1166-4fd5-91be-3ae43534d151","order_by":2,"name":"Huaiping Zhu","email":"","orcid":"","institution":"York University","correspondingAuthor":false,"prefix":"","firstName":"Huaiping","middleName":"","lastName":"Zhu","suffix":""}],"badges":[],"createdAt":"2025-06-09 14:53:24","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6855591/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6855591/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":89862713,"identity":"a424cf52-0b94-49a0-8efa-75e8af9a2c9b","added_by":"auto","created_at":"2025-08-25 21:31:25","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":891910,"visible":true,"origin":"","legend":"","description":"","filename":"OPMLcancerimmunotherapy.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6855591/v1_covered_7f4d6d0a-9eea-4641-9271-800a5bd6997e.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Universal differential equations for optimal control problems and its application on cancer therapy","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":"Optimal control, scientific machine learning, universal differential equations, cancer therapy","lastPublishedDoi":"10.21203/rs.3.rs-6855591/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6855591/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This paper highlights a parallel between the forward–backward sweeping method for optimal control and deep learning training procedures. 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