A Neural-Network Method for Solving the Optimal Control Model of Carbon Emissions

A Neural-Network Method for Solving the Optimal Control Model of Carbon Emissions | 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 Neural-Network Method for Solving the Optimal Control Model of Carbon Emissions Yihan Zhou, Chun Jiang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7624820/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 China has pledged to reach carbon neutrality by 2060 and peak its carbon emissions before 2030, making carbon emission control a pressing issue in environmental economics. In the Yangtze River Delta (YRD), optimal carbon-emission control poses significant computational challenges due to high dimensionality, nonlinear dynamics, and complex boundary conditions. Traditional solution methods—including linear–quadratic regulators, finite-difference Hamilton–Jacobi–Bellman (HJB) solvers, spectral collocation, and reinforcement-learning–based approaches—have succeeded in low-dimensional or simplified models but often struggle with scalability and accurate enforcement of boundary and terminal constraints. Recent progress in triangular basis neural networks (TBNNs) highlights their ability to embed partial differential equation (PDE) residuals into network training and overcome these limitations. Therefore, we propose a unified residual-based collocation TBNN framework to solve the high-dimensional HJB equation governing stochastic carbon-emission control in Shanghai, Jiangsu, and Zhejiang. Our network parameterizes the value function in (t, I) space, and training minimizes a composite loss comprising the squared HJB residual, boundary-condition penalties, and terminal-condition error, using stochastic mini-batches and adaptive learning rates to ensure stable, rapid convergence without grid discretization. Applied to 2000–2022 regional emissions, GDP, and population data, the TBNN yields value-function surfaces bounded within approximately ±10⁵, exhibits sharp peaks at high emission intensities, and captures subtle temporal undulations without clear trends. Simultaneously, the cross-entropy loss remains below 10¹, falling under 0.1 after 5,000 epochs, which reflects high signal-to-noise fidelity. From these surfaces we derive actionable, region-specific investment guidelines—CNY 20–60/ton for Jiangsu, CNY 50–120/ton for Shanghai, and CNY 250–650/ton for Zhejiang—providing a robust, data-driven tool for real-time, adaptive emission-reduction planning across heterogeneous regional dynamics. Hamilton–Jacobi–Bellman Equation Triangular Basis Neural Network Carbon Emission Control Yangtze River Delta Residual-Based Collocation Full Text Additional Declarations The authors declare no competing interests. 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. 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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-7624820","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":515611854,"identity":"84252bed-db3f-40bd-9262-43538cdf59c2","order_by":0,"name":"Yihan Zhou","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABFElEQVRIie3PsUrDQBjA8e8MZLq0m1w9aHyElEBALOmrJBxkCj6BQ0Iho3OlFl/B0THl4FyiWQ8iWBBcTCEu0km8UsElwboJ3h/u4OPuN3wAOt2fjqiDkrG6jQQg359EACb6HeE/E+funr/i80f/ej4V1Loth85Tmq6aAuz+YbtzirPoFIsXll6JiFpF5ToCTUczCaPLedBO8thzscmZQWKPWlkV3giUUdxA4FQdpKwV+eDM3JGHPYiM3Wcr4z7ekfyLyG4ykLWHFhc8ICRiJ4uMuQMRql0K0rlLr4zdpn7nE3vGlnKd+cMe58tVI8Z2n7aT4xzMIwwQJmo4wN8PpPX7NjsB420DMNkOaNP5T6fT6f5zn1XSZ18aNofVAAAAAElFTkSuQmCC","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Yihan","middleName":"","lastName":"Zhou","suffix":""},{"id":515611982,"identity":"266e35ac-1ec4-490f-8385-c2efda0a1fc5","order_by":1,"name":"Chun Jiang","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Chun","middleName":"","lastName":"Jiang","suffix":""}],"badges":[],"createdAt":"2025-09-16 01:59:29","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-7624820/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7624820/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":91495380,"identity":"d3459851-c5d0-4a7e-a8aa-fe03c79c75bc","added_by":"auto","created_at":"2025-09-17 06:13:42","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1652217,"visible":true,"origin":"","legend":"","description":"","filename":"submission.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7624820/v1_covered_cbab33e2-1a8a-4322-8dff-2b708ab8d8e4.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eA Neural-Network Method for Solving the Optimal Control Model of Carbon Emissions\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"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":"Hamilton–Jacobi–Bellman Equation, Triangular Basis Neural Network, Carbon Emission Control, Yangtze River Delta, Residual-Based Collocation","lastPublishedDoi":"10.21203/rs.3.rs-7624820/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7624820/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eChina has pledged to reach carbon neutrality by 2060 and peak its carbon emissions before 2030, making carbon emission control a pressing issue in environmental economics. In the Yangtze River Delta (YRD), optimal carbon-emission control poses significant computational challenges due to high dimensionality, nonlinear dynamics, and complex boundary conditions. Traditional solution methods—including linear–quadratic regulators, finite-difference \u003cstrong\u003eHamilton–Jacobi–Bellman (HJB)\u003c/strong\u003e solvers, spectral collocation, and reinforcement-learning–based approaches—have succeeded in low-dimensional or simplified models but often struggle with scalability and accurate enforcement of boundary and terminal constraints. Recent progress in \u003cstrong\u003etriangular basis neural networks (TBNNs)\u003c/strong\u003e highlights their ability to embed partial differential equation (PDE) residuals into network training and overcome these limitations.\u003c/p\u003e\n\u003cp\u003eTherefore, we propose a unified \u003cstrong\u003eresidual-based collocation TBNN\u003c/strong\u003e framework to solve the high-dimensional HJB equation governing stochastic carbon-emission control in Shanghai, Jiangsu, and Zhejiang. Our network parameterizes the value function in \u003cem\u003e(t, I)\u003c/em\u003e space, and training minimizes a composite loss comprising the squared HJB residual, boundary-condition penalties, and terminal-condition error, using stochastic mini-batches and adaptive learning rates to ensure stable, rapid convergence without grid discretization.\u003c/p\u003e\n\u003cp\u003eApplied to 2000–2022 regional emissions, GDP, and population data, the TBNN yields value-function surfaces bounded within approximately ±10⁵, exhibits sharp peaks at high emission intensities, and captures subtle temporal undulations without clear trends. Simultaneously, the cross-entropy loss remains below 10¹, falling under 0.1 after 5,000 epochs, which reflects high signal-to-noise fidelity. 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