mlsanityData to 14 Oct 2026

A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems

arXiv 1912.01825 ↗First posted 4 Dec 2019cs.LG cs.NA math.NAcited by 97

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Abstract (arXiv)

Mean field games (MFG) and mean field control (MFC) are critical classes of multi-agent models for efficient analysis of massive populations of interacting agents. Their areas of application span topics in economics, finance, game theory, industrial engineering, crowd motion, and more. In this paper, we provide a flexible machine learning framework for the numerical solution of potential MFG and MFC models. State-of-the-art numerical methods for solving such problems utilize spatial discretization that leads to a curse-of-dimensionality. We approximately solve high-dimensional problems by combining Lagrangian and Eulerian viewpoints and leveraging recent advances from machine learning. More precisely, we work with a Lagrangian formulation of the problem and enforce the underlying Hamilton-Jacobi-Bellman (HJB) equation that is derived from the Eulerian formulation. Finally, a tailored neural network parameterization of the MFG/MFC solution helps us avoid any spatial discretization. Our numerical results include the approximate solution of 100-dimensional instances of optimal transport and crowd motion problems on a standard work station and a validation using an Eulerian solver in two dimensions. These results open the door to much-anticipated applications of MFG and MFC models that were beyond reach with existing numerical methods.

Key facts. Cited by 97 papers in the map (data through 14 Oct 2026). First posted on arXiv 4 Dec 2019.

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 15; 2022: 12; 2023: 17; 2024: 17; 2025: 21; 2026: 15.

97 papers.

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  1. 2021An Introduction to Deep Generative Modelingcited by 104
  2. 2023In-Context Operator Learning with Data Prompts for Differential Equation Problemscited by 101
  3. 2021JFB: Jacobian-Free Backpropagation for Implicit Networkscited by 86
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  7. 2022Normalizing flow neural networks by JKO schemecited by 39
  8. 2023Convergence of flow-based generative models via proximal gradient descent in Wasserstein spacecited by 34
  9. 2024Learning stochastic dynamics from snapshots through regularized unbalanced optimal transportcited by 33
  10. 2021Scaling up Mean Field Games with Online Mirror Descentcited by 32
  11. 2021Mean Field Games Flock! The Reinforcement Learning Waycited by 31
  12. 2024PDE Generalization of In-Context Operator Networks: A Study on 1D Scalar Nonlinear Conservation Lawscited by 31
38 more
  1. 2023A mean-field games laboratory for generative modelingcited by 28
  2. 2021On the Representation of Solutions to Elliptic PDEs in Barron Spacescited by 25
  3. 2023An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensionscited by 21
  4. 2021A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problemscited by 20
  5. 2021Deep Learning for Mean Field Games and Mean Field Control with Applications to Financecited by 20
  6. 2023Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inferencecited by 20
  7. 2024Dynamical Measure Transport and Neural PDE Solvers for Samplingcited by 20
  8. 2025Joint Velocity-Growth Flow Matching for Single-Cell Dynamics Modelingcited by 20
  9. 2025From discrete-time policies to continuous-time diffusion samplers: Asymptotic equivalences and faster trainingcited by 19
  10. 2022Bridging Mean-Field Games and Normalizing Flows with Trajectory Regularizationcited by 17
  11. 2022Taming Hyperparameter Tuning in Continuous Normalizing Flows Using the JKO Schemecited by 17
  12. 2022Mean-field neural networks: learning mappings on Wasserstein spacecited by 16
  13. 2024Neural Optimal Transport with Lagrangian Costscited by 16
  14. 2021Reinforcement Learning for Mean Field Games, with Applications to Economicscited by 15
  15. 2022Explainable AI via Learning to Optimizecited by 14
  16. 2021ResNet-LDDMM: Advancing the LDDMM Framework using Deep Residual Networkscited by 13
  17. 2022Learning High-Dimensional McKean-Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependencecited by 13
  18. 2021Signatured Deep Fictitious Play for Mean Field Games with Common Noisecited by 12
  19. 2025Modeling Cell Dynamics and Interactions with Unbalanced Mean Field Schr"odinger Bridgecited by 12
  20. 2024Annealing Flow Generative Models Towards Sampling High-Dimensional and Multi-Modal Distributionscited by 11
  21. 2021Learning High Dimensional Wasserstein Geodesicscited by 10
  22. 2022Mean-field neural networks-based algorithms for McKean-Vlasov control problems *cited by 10
  23. 2023A Policy Gradient Framework for Stochastic Optimal Control Problems with Global Convergence Guaranteecited by 10
  24. 2021DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle methodcited by 9
  25. 2025Integrating Dynamical Systems Modeling with Spatiotemporal scRNA-seq Data Analysiscited by 9
  26. 2023Computing high-dimensional optimal transport by flow neural networkscited by 8
  27. 2023RDA-INR: Riemannian Diffeomorphic Autoencoding via Implicit Neural Representationscited by 7
  28. 2023Actor critic learning algorithms for mean-field control with moment neural networkscited by 7
  29. 2024Differential Equations for Continuous-Time Deep Learningcited by 7
  30. 2024On Bellman equations for continuous-time policy evaluation I: discretization and approximationcited by 7
  31. 2024Score-based Neural Ordinary Differential Equations for Computing Mean Field Control Problemscited by 7
  32. 2024Deep Backward and Galerkin Methods for the Finite State Master Equationcited by 6
  33. 2022A scalable deep learning approach for solving high-dimensional dynamic optimal transportcited by 5
  34. 2024Go With the Flow: Fast Diffusion for Gaussian Mixture Modelscited by 5
  35. 2022Tangential Wasserstein Projectionscited by 4
  36. 2022On the Near-Optimality of Local Policies in Large Cooperative Multi-Agent Reinforcement Learningcited by 4
  37. 2023Fine-Tune Language Models as Multi-Modal Differential Equation Solverscited by 4
  38. 2024Neural McKean-Vlasov Processes: Distributional Dependence in Diffusion Processescited by 4

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