mlsanityData to 14 Oct 2026

Score-based Neural Ordinary Differential Equations for Computing Mean Field Control Problems

arXiv 2409.16471 ↗First posted 24 Sep 2024math.OC cs.LGcited by 7

Not on the map. Why?

Abstract (arXiv)

Classical neural ordinary differential equations (ODEs) are powerful tools for approximating the log-density functions in high-dimensional spaces along trajectories, where neural networks parameterize the velocity fields. This paper proposes a system of neural differential equations representing first- and second-order score functions along trajectories based on deep neural networks. We reformulate the mean field control (MFC) problem with individual noises into an unconstrained optimization problem framed by the proposed neural ODE system. Additionally, we introduce a novel regularization term to enforce characteristics of viscous Hamilton--Jacobi--Bellman (HJB) equations to be satisfied based on the evolution of the second-order score function. Examples include regularized Wasserstein proximal operators (RWPOs), probability flow matching of Fokker--Planck (FP) equations, and linear quadratic (LQ) MFC problems, which demonstrate the effectiveness and accuracy of the proposed method.

Key facts. Cited by 7 papers in the map (data through 14 Oct 2026). Builds on 39 of its 58 references that are in the map. First posted on arXiv 24 Sep 2024.

Builds on

39 of 58 references.

References · 39

  1. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 9,122
  2. 2022Flow Matching for Generative Modelingcited by 5,702
  3. 2018Neural Ordinary Differential Equationscited by 3,779
  4. 2015Variational Inference with Normalizing Flowscited by 2,170
  5. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,323
  6. 2019Augmented Neural ODEscited by 397
  7. 2017Stable Architectures for Deep Neural Networkscited by 341
  8. 2023The probability flow ODE is provably fastcited by 146
  9. 2017A Tutorial on Kernel Density Estimation and Recent Advancescited by 101
  10. 2019A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problemscited by 97
  11. 2022Probability flow solution of the Fokker-Planck equationcited by 47
  12. 2023A Computational Framework for Solving Wasserstein Lagrangian Flowscited by 46
27 more
  1. 2019Accelerated Information Gradient flowcited by 39
  2. 2022Normalizing flow neural networks by JKO schemecited by 39
  3. 2020Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approachcited by 35
  4. 2020Alternating the Population and Control Neural Networks to Solve High-Dimensional Stochastic Mean-Field Gamescited by 35
  5. 2019Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: II -- The Finite Horizon Casecited by 34
  6. 2021Actor-Critic Method for High Dimensional Static Hamilton--Jacobi--Bellman Partial Differential Equations based on Neural Networkscited by 29
  7. 2023A mean-field games laboratory for generative modelingcited by 28
  8. 2023Recent Developments in Machine Learning Methods for Stochastic Control and Gamescited by 24
  9. 2023Actor-Critic learning for mean-field control in continuous timecited by 22
  10. 2022Self-Consistency of the Fokker-Planck Equationcited by 18
  11. 2022Bridging Mean-Field Games and Normalizing Flows with Trajectory Regularizationcited by 17
  12. 2022Taming Hyperparameter Tuning in Continuous Normalizing Flows Using the JKO Schemecited by 17
  13. 2023Score-based Transport Modeling for Mean-Field Fokker-Planck Equationscited by 14
  14. 2016Limit theory for controlled McKean-Vlasov dynamicscited by 12
  15. 2023A kernel formula for regularized Wasserstein proximal operatorscited by 11
  16. 2021A fast iterative PDE-based algorithm for feedback controls of nonsmooth mean-field control problemscited by 10
  17. 2023A Policy Gradient Framework for Stochastic Optimal Control Problems with Global Convergence Guaranteecited by 10
  18. 2013Mean-Field Optimal Controlcited by 9
  19. 2022Optimal Neural Network Approximation of Wasserstein Gradient Direction via Convex Optimizationcited by 9
  20. 2022A Neural Network Warm-Start Approach for the Inverse Acoustic Obstacle Scattering Problemcited by 8
  21. 2024Solving Time-Continuous Stochastic Optimal Control Problems: Algorithm Design and Convergence Analysis of Actor-Critic Flowcited by 8
  22. 2023Noise-Free Sampling Algorithms via Regularized Wasserstein Proximalscited by 7
  23. 2019Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: I -- The Ergodic Casecited by 6
  24. 2024A deep learning algorithm for computing mean field control problems via forward-backward score dynamicscited by 5
  25. 2018On the convergence of closed-loop Nash equilibria to the mean field game limitcited by 4
  26. 2021Mean Field Games Master Equations with Non-separable Hamiltonians and Displacement Monotonicitycited by 4
  27. 2023Deep Learning for Population-Dependent Controls in Mean Field Control Problems with Common Noisecited by 4

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 1; 2025: 4; 2026: 2.

7 papers.

  1. 2024Learning stochastic dynamics from snapshots through regularized unbalanced optimal transportcited by 33
  2. 2025Modeling Cell Dynamics and Interactions with Unbalanced Mean Field Schr"odinger Bridgecited by 12
  3. 2025Integrating Dynamical Systems Modeling with Spatiotemporal scRNA-seq Data Analysiscited by 9
  4. 2026Flow Learners for PDEs: Toward a Physics-to-Physics Paradigm for Scientific Computingcited by 4
  5. 2025Neural Hamilton--Jacobi Characteristic Flows for Optimal Transportcited by 1
  6. 2025Divergence-kernel method for linear responses of densities and generative modelscited by 0
  7. 2026Self-supervised In-context Operator Learning for Stochastic Mean-Field Controlcited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2025Neural Actor-Critic Methods for Hamilton-Jacobi-Bellman PDEs: Asymptotic Analysis and Numerical Studies
  2. 2026Autoregression-Free Neural Operators for Time-Dependent PDEs
  3. 2022Neural Ordinary Differential Equations for Nonlinear System Identification
  4. 2025CFO: Learning Continuous-Time PDE Dynamics via Flow-Matched Neural Operators
  5. 2024Simulation-Free Training of Neural ODEs on Paired Data
  6. 2026Stability and Discretization Error of State Space Model Neural Operators
  7. 2023Noise in the reverse process improves the approximation capabilities of diffusion models
  8. 2024From Fourier to Neural ODEs: Flow Matching for Modeling Complex Systems