Solving Nonlinear and Complex Optimal Control Problems via Multi-task Artificial Neural Networks

Solving Nonlinear and Complex Optimal Control Problems via Multi-task Artificial Neural Networks | 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 Article Solving Nonlinear and Complex Optimal Control Problems via Multi-task Artificial Neural Networks Alaeddin Malek, Ali Emami Kerdabadi This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6134840/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 14 Jul, 2025 Read the published version in Scientific Reports → Version 1 posted 10 You are reading this latest preprint version Abstract This article proposes a novel approach using multi-task learning for solving nonlinear and complex optimal control problems. A neural network-based framework is proposed by unifying state, control, and adjoint dynamics into the three separate neural networks. A specifc structure is designed to embed the Hamiltonian into a neural network framework for solving optimal control problems using the Pontryagin Maximum Principle. An iterative algorithm that synergizes specifc structures is proposed for neural network learning sequentially and parallel. It is proved that the solution of neural networks converges to the main optimal control problem solution. This ensures that the Hamiltonian optimality condition is satisfed. To evaluate the current approach, two nonlinear complex optimal control problems in the feld of epidemiology and power grid stabilization are solved successfully. Numerical results are given, and related graphs are depicted. Biological sciences/Computational biology and bioinformatics/Computational models Biological sciences/Computational biology and bioinformatics/Machine learning Health sciences/Diseases Physical sciences/Mathematics and computing/Applied mathematics Physical sciences/Mathematics and computing/Computational science Full Text Additional Declarations No competing interests reported. Supplementary Files Supplementary.pdf Cite Share Download PDF Status: Published Journal Publication published 14 Jul, 2025 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 29 May, 2025 Reviews received at journal 26 May, 2025 Reviews received at journal 06 May, 2025 Reviewers agreed at journal 05 May, 2025 Reviewers agreed at journal 05 May, 2025 Reviewers invited by journal 05 May, 2025 Editor assigned by journal 05 May, 2025 Editor invited by journal 05 May, 2025 Submission checks completed at journal 05 May, 2025 First submitted to journal 01 Mar, 2025 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-6134840","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":452665295,"identity":"4a6937c6-a5eb-40ef-b406-702b1e8ad142","order_by":0,"name":"Alaeddin Malek","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyUlEQVRIiWNgGAWjYFAC5gaGDwwMjA0wPhthLYwNjDNI1sLMg6yFINBtP9j42TbHRrZ/9uHDHxhq7Bj4pA/g12J2JrFZOndbmvGMc2lpEgzHkhnY+BIIaDmQ2ADUcjix4QyPGdAjBxjYeAg4zOz8w+bflkAt88/wf/7A8I8YLTcS26QZgVo2nOFhkGBsI0rLwzbLXqBfNp5hM5NI7EvmIcJhyYdv/NxmIzvvDPPjDx++2cnJ9xDQggoSGBgI2TEKRsEoGAWjgBgAAKKOQRUB2dqWAAAAAElFTkSuQmCC","orcid":"","institution":"Tarbiat Modares University","correspondingAuthor":true,"prefix":"","firstName":"Alaeddin","middleName":"","lastName":"Malek","suffix":""},{"id":452665296,"identity":"4f954ce4-bdc1-4001-9126-3b131b17de94","order_by":1,"name":"Ali Emami Kerdabadi","email":"","orcid":"","institution":"Tarbiat Modares University","correspondingAuthor":false,"prefix":"","firstName":"Ali","middleName":"Emami","lastName":"Kerdabadi","suffix":""}],"badges":[],"createdAt":"2025-03-01 13:08:07","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6134840/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6134840/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41598-025-10339-w","type":"published","date":"2025-07-14T16:05:29+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":87220469,"identity":"7d7cf28a-95c5-4717-ad95-2391ee6e38da","added_by":"auto","created_at":"2025-07-21 16:12:30","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":7190500,"visible":true,"origin":"","legend":"","description":"","filename":"EmamiMalekSRs1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6134840/v1_covered_9814f7bc-40b5-4e21-bb9d-eb9aa44c443e.pdf"},{"id":82323153,"identity":"a5dcadb1-c81f-419d-901a-5f2b3100e6f6","added_by":"auto","created_at":"2025-05-09 05:41:48","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":72766,"visible":true,"origin":"","legend":"","description":"","filename":"Supplementary.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6134840/v1/65ad5dbbfa657da8b893415d.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Solving Nonlinear and Complex Optimal Control Problems via Multi-task Artificial Neural Networks","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":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-6134840/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6134840/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This article proposes a novel approach using multi-task learning for solving nonlinear and complex optimal control problems. A neural network-based framework is proposed by unifying state, control, and adjoint dynamics into the three separate neural networks. A specifc structure is designed to embed the Hamiltonian into a neural network framework for solving optimal control problems using the Pontryagin Maximum Principle. An iterative algorithm that synergizes specifc structures is proposed for neural network learning sequentially and parallel. It is proved that the solution of neural networks converges to the main optimal control problem solution. This ensures that the Hamiltonian optimality condition is satisfed. To evaluate the current approach, two nonlinear complex optimal control problems in the feld of epidemiology and power grid stabilization are solved successfully. Numerical results are given, and related graphs are depicted.","manuscriptTitle":"Solving Nonlinear and Complex Optimal Control Problems via Multi-task Artificial Neural Networks","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-05-09 05:41:43","doi":"10.21203/rs.3.rs-6134840/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-05-30T03:25:24+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-05-27T01:24:00+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-05-06T12:13:39+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"269348019551906445555890731989727709538","date":"2025-05-06T03:25:09+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"83000340277838079556999717636097277510","date":"2025-05-06T02:54:37+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-05-06T02:49:05+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-05-06T02:47:34+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2025-05-05T11:27:24+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-05-05T11:24:38+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2025-03-01T12:53:31+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"1cf5335d-f8b3-4b08-8f7d-49b37ed35d33","owner":[],"postedDate":"May 9th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":48139993,"name":"Biological sciences/Computational biology and bioinformatics/Computational models"},{"id":48139994,"name":"Biological sciences/Computational biology and bioinformatics/Machine learning"},{"id":48139995,"name":"Health sciences/Diseases"},{"id":48139996,"name":"Physical sciences/Mathematics and computing/Applied mathematics"},{"id":48139997,"name":"Physical sciences/Mathematics and computing/Computational science"}],"tags":[],"updatedAt":"2025-07-21T16:10:27+00:00","versionOfRecord":{"articleIdentity":"rs-6134840","link":"https://doi.org/10.1038/s41598-025-10339-w","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2025-07-14 16:05:29","publishedOnDateReadable":"July 14th, 2025"},"versionCreatedAt":"2025-05-09 05:41:43","video":"","vorDoi":"10.1038/s41598-025-10339-w","vorDoiUrl":"https://doi.org/10.1038/s41598-025-10339-w","workflowStages":[]},"version":"v1","identity":"rs-6134840","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6134840","identity":"rs-6134840","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}