Initialization-driven neural generation and training for high-dimensional optimal control and first-order mean field games

[2507.15126] Initialization-driven neural generation and training for high-dimensional optimal control and first-order mean field games Skip to main content Search Submit Donate Log in Mathematics > Optimization and Control arXiv:2507.15126 (math) [Submitted on 20 Jul 2025 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: Initialization-driven neural generation and training for high-dimensional optimal control and first-order mean field games Authors: Mouhcine Assouli , Justina Gianatti , Badr Missaoui , Francisco J. Silva View PDF HTML (experimental) Abstract: This paper first introduces a method to approximate the value function of high-dimensional optimal control by neural networks. Based on the established relationship between Pontryagin's maximum principle (PMP) and the value function of the optimal control problem, which is characterized as being the unique solution to an associated Hamilton-Jacobi-Bellman (HJB) equation, we propose an approach that begins by using neural networks to provide a first rough estimate of the value function, which serves as initialization for solving the two point boundary value problem in the PMP and, as a result, generates reliable data. To train the neural network we define a loss function that takes into account this dataset and also penalizes deviations from the HJB equation. In the second part, we address the computation of equilibria in first-order Mean Field Game (MFG) problems by integrating our method with the fictitious play algorithm. These equilibria are characterized by a coupled system of a first-order HJB equation and a continuity equation. To approximate the solution to the continuity equation, we introduce a second neural network that learns the flow map transporting the initial distribution of agents. This network is trained on data generated by solving the underlying ODEs for a batch of initial conditions sampled from the initial distribution of agents. By combining this flow approximation, the previously described method for approximating the value function, and the fictitious play algorithm, we obtain an effective method to tackle high-dimensional deterministic MFGs. Comments: 24 pages, 8 figures Subjects: Optimization and Control (math.OC) MSC classes: 49N70, 35F21, 91A13, 68T05, 68Q32 Cite as: arXiv:2507.15126 [math.OC] (or arXiv:2507.15126v2 [math.OC] for this version) https://doi.org/10.48550/arXiv.2507.15126 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Mouhcine Assouli [ view email ] [v1] Sun, 20 Jul 2025 21:30:23 UTC (6,265 KB) [v2] Thu, 27 Aug 2026 10:21:36 UTC (3,556 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.OC < prev | next > new | recent | 2025-07 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

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