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

arXiv: https://arxiv.org/abs/2409.16471 · HTML: https://mlsanity.com/paper/2409.16471/

## 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 (most cited first, 30 of 39)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023)
- [A Tutorial on Kernel Density Estimation and Recent Advances](https://mlsanity.com/paper/1704.03924/) (2017)
- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [Probability flow solution of the Fokker-Planck equation](https://mlsanity.com/paper/2206.04642/) (2022)
- [A Computational Framework for Solving Wasserstein Lagrangian Flows](https://mlsanity.com/paper/2310.10649/) (2023)
- [Accelerated Information Gradient flow](https://mlsanity.com/paper/1909.02102/) (2019)
- [Normalizing flow neural networks by JKO scheme](https://mlsanity.com/paper/2212.14424/) (2022)
- [Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach](https://mlsanity.com/paper/2002.02600/) (2020)
- [Alternating the Population and Control Neural Networks to Solve High-Dimensional Stochastic Mean-Field Games](https://mlsanity.com/paper/2002.10113/) (2020)
- [Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: II -- The Finite Horizon Case](https://mlsanity.com/paper/1908.01613/) (2019)
- [Actor-Critic Method for High Dimensional Static Hamilton--Jacobi--Bellman Partial Differential Equations based on Neural Networks](https://mlsanity.com/paper/2102.11379/) (2021)
- [A mean-field games laboratory for generative modeling](https://mlsanity.com/paper/2304.13534/) (2023)
- [Recent Developments in Machine Learning Methods for Stochastic Control and Games](https://mlsanity.com/paper/2303.10257/) (2023)
- [Actor-Critic learning for mean-field control in continuous time](https://mlsanity.com/paper/2303.06993/) (2023)
- [Self-Consistency of the Fokker-Planck Equation](https://mlsanity.com/paper/2206.00860/) (2022)
- [Bridging Mean-Field Games and Normalizing Flows with Trajectory Regularization](https://mlsanity.com/paper/2206.14990/) (2022)
- [Taming Hyperparameter Tuning in Continuous Normalizing Flows Using the JKO Scheme](https://mlsanity.com/paper/2211.16757/) (2022)
- [Score-based Transport Modeling for Mean-Field Fokker-Planck Equations](https://mlsanity.com/paper/2305.03729/) (2023)
- [Limit theory for controlled McKean-Vlasov dynamics](https://mlsanity.com/paper/1609.08064/) (2016)
- [A kernel formula for regularized Wasserstein proximal operators](https://mlsanity.com/paper/2301.10301/) (2023)
- [A fast iterative PDE-based algorithm for feedback controls of nonsmooth mean-field control problems](https://mlsanity.com/paper/2108.06740/) (2021)
- [A Policy Gradient Framework for Stochastic Optimal Control Problems with Global Convergence Guarantee](https://mlsanity.com/paper/2302.05816/) (2023)
- [Mean-Field Optimal Control](https://mlsanity.com/paper/1306.5913/) (2013)

## Built on by (most cited first, 7 of 7)

- [Learning stochastic dynamics from snapshots through regularized unbalanced optimal transport](https://mlsanity.com/paper/2410.00844/) (2024)
- [Modeling Cell Dynamics and Interactions with Unbalanced Mean Field Schr"odinger Bridge](https://mlsanity.com/paper/2505.11197/) (2025)
- [Integrating Dynamical Systems Modeling with Spatiotemporal scRNA-seq Data Analysis](https://mlsanity.com/paper/2503.11347/) (2025)
- [Flow Learners for PDEs: Toward a Physics-to-Physics Paradigm for Scientific Computing](https://mlsanity.com/paper/2604.07366/) (2026)
- [Neural Hamilton--Jacobi Characteristic Flows for Optimal Transport](https://mlsanity.com/paper/2510.01153/) (2025)
- [Divergence-kernel method for linear responses of densities and generative models](https://mlsanity.com/paper/2509.03992/) (2025)
- [Self-supervised In-context Operator Learning for Stochastic Mean-Field Control](https://mlsanity.com/paper/2608.18282/) (2026)

## 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.