# Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: II -- The Finite Horizon Case

arXiv: https://arxiv.org/abs/1908.01613 · HTML: https://mlsanity.com/paper/1908.01613/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Aug 2019.

## Built on by (most cited first, 30 of 34)

- [Deep Generalized Schr"odinger Bridge](https://mlsanity.com/paper/2209.09893/) (2022)
- [Stochastic Optimal Control Matching](https://mlsanity.com/paper/2312.02027/) (2023)
- [Scalable Deep Reinforcement Learning Algorithms for Mean Field Games](https://mlsanity.com/paper/2203.11973/) (2022)
- [Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality](https://mlsanity.com/paper/2106.08900/) (2021)
- [Scaling up Mean Field Games with Online Mirror Descent](https://mlsanity.com/paper/2103.00623/) (2021)
- [Generalization in Mean Field Games by Learning Master Policies](https://mlsanity.com/paper/2109.09717/) (2021)
- [Recent Developments in Machine Learning Methods for Stochastic Control and Games](https://mlsanity.com/paper/2303.10257/) (2023)
- [Deep Learning for Mean Field Games and Mean Field Control with Applications to Finance](https://mlsanity.com/paper/2107.04568/) (2021)
- [Reinforcement Learning for Mean Field Games, with Applications to Economics](https://mlsanity.com/paper/2106.13755/) (2021)
- [Signatured Deep Fictitious Play for Mean Field Games with Common Noise](https://mlsanity.com/paper/2106.03272/) (2021)
- [A Policy Gradient Framework for Stochastic Optimal Control Problems with Global Convergence Guarantee](https://mlsanity.com/paper/2302.05816/) (2023)
- [Score-based Neural Ordinary Differential Equations for Computing Mean Field Control Problems](https://mlsanity.com/paper/2409.16471/) (2024)
- [Deep Backward and Galerkin Methods for the Finite State Master Equation](https://mlsanity.com/paper/2403.04975/) (2024)
- [Deep Learning for Principal-Agent Mean Field Games](https://mlsanity.com/paper/2110.01127/) (2021)
- [A brief review of the Deep BSDE method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2505.17032/) (2025)
- [A Forward Propagation Algorithm for Online Optimization of Nonlinear Stochastic Differential Equations](https://mlsanity.com/paper/2207.04496/) (2022)
- [Exploration noise for learning linear-quadratic mean field games](https://mlsanity.com/paper/2107.00839/) (2021)
- [Simultaneously Solving Infinitely Many LQ Mean Field Games In Hilbert Spaces: The Power of Neural Operators](https://mlsanity.com/paper/2510.20017/) (2025)
- [Langevin algorithms for Markovian Neural Networks and Deep Stochastic control](https://mlsanity.com/paper/2212.12018/) (2022)
- [Decoding Mean Field Games from Population and Environment Observations By Gaussian Processes](https://mlsanity.com/paper/2312.06625/) (2023)
- [Convergence of the Deep Galerkin Method for Finite State Mean Field Control Problems](https://mlsanity.com/paper/2405.13346/) (2024)
- [Learning Surrogate Potential Mean Field Games via Gaussian Processes: A Data-Driven Approach to Ill-Posed Inverse Problems](https://mlsanity.com/paper/2502.11506/) (2025)
- [Finite-Sample Convergence Bounds for Trust Region Policy Optimization in Mean-Field Games](https://mlsanity.com/paper/2505.22781/) (2025)
- [Operator Learning for Families of Finite-State Mean-Field Games](https://mlsanity.com/paper/2602.13169/) (2026)
- [Deep Learning for Mean Field Optimal Transport](https://mlsanity.com/paper/2302.14739/) (2023)
- [Policy Gradient Optimal Correlation Search for Variance Reduction in Monte Carlo simulation and Maximum Optimal Transport](https://mlsanity.com/paper/2307.12703/) (2023)
- [Learning to Stop: Deep Learning for Mean Field Optimal Stopping](https://mlsanity.com/paper/2410.08850/) (2024)
- [Kernel-based potential mean-field games with unbiased random Fourier U-statistics](https://mlsanity.com/paper/2605.29371/) (2026)
- [Stability Via Adversarial Training of Neural Network Stochastic Control of Mean-Field Type](https://mlsanity.com/paper/2210.00874/) (2022)
- [A Deep Learning Method for Optimal Investment Under Relative Performance Criteria Among Heterogeneous Agents](https://mlsanity.com/paper/2402.07365/) (2024)

## Abstract (arXiv)

We propose two numerical methods for the optimal control of McKean-Vlasov dynamics in finite time horizon. Both methods are based on the introduction of a suitable loss function defined over the parameters of a neural network. This allows the use of machine learning tools, and efficient implementations of stochastic gradient descent in order to perform the optimization. In the first method, the loss function stems directly from the optimal control problem. The second method tackles a generic forward-backward stochastic differential equation system (FBSDE) of McKean-Vlasov type, and relies on suitable reformulation as a mean field control problem. To provide a guarantee on how our numerical schemes approximate the solution of the original mean field control problem, we introduce a new optimization problem, directly amenable to numerical computation, and for which we rigorously provide an error rate. Several numerical examples are provided. Both methods can easily be applied to certain problems with common noise, which is not the case with the existing technology. Furthermore, although the first approach is designed for mean field control problems, the second is more general and can also be applied to the FBSDE arising in the theory of mean field games.