# Convergence of the Deep Galerkin Method for Finite State Mean Field Control Problems

arXiv: https://arxiv.org/abs/2405.13346 · HTML: https://mlsanity.com/paper/2405.13346/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 58 references that are in the map.
- First posted on arXiv 22 May 2024.

## Builds on (most cited first, 20 of 20)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Challenges in Training PINNs: A Loss Landscape Perspective](https://mlsanity.com/paper/2402.01868/) (2024)
- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [On the approximation of functions by tanh neural networks](https://mlsanity.com/paper/2104.08938/) (2021)
- [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)
- [Minimizing the Maximal Loss: How and Why?](https://mlsanity.com/paper/1602.01690/) (2016)
- [Actor-Critic Method for High Dimensional Static Hamilton--Jacobi--Bellman Partial Differential Equations based on Neural Networks](https://mlsanity.com/paper/2102.11379/) (2021)
- [Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations](https://mlsanity.com/paper/2206.09527/) (2022)
- [Deep Learning Methods for Mean Field Control Problems with Delay](https://mlsanity.com/paper/1905.00358/) (2019)
- [Mean-field Markov decision processes with common noise and open-loop controls](https://mlsanity.com/paper/1912.07883/) (2019)
- [Deep Learning for Mean Field Games and Mean Field Control with Applications to Finance](https://mlsanity.com/paper/2107.04568/) (2021)
- [Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics](https://mlsanity.com/paper/1604.04057/) (2016)
- [Limit theory for controlled McKean-Vlasov dynamics](https://mlsanity.com/paper/1609.08064/) (2016)
- [Bellman equation and viscosity solutions for mean-field stochastic control problem](https://mlsanity.com/paper/1512.07866/) (2015)
- [Global Convergence of Deep Galerkin and PINNs Methods for Solving Partial Differential Equations](https://mlsanity.com/paper/2305.06000/) (2023)
- [Partially Observable Mean Field Multi-Agent Reinforcement Learning Based on Graph-Attention](https://mlsanity.com/paper/2304.12653/) (2023)
- [Deep Backward and Galerkin Methods for the Finite State Master Equation](https://mlsanity.com/paper/2403.04975/) (2024)
- [Incentivizing Sharing in Realtime D2D Streaming Networks: A Mean Field Game Perspective](https://mlsanity.com/paper/1604.02435/) (2016)
- [Finite state N-agent and mean field control problems](https://mlsanity.com/paper/2010.11569/) (2020)

## Built on by (most cited first, 2 of 2)

- [Neural Actor-Critic Methods for Hamilton-Jacobi-Bellman PDEs: Asymptotic Analysis and Numerical Studies](https://mlsanity.com/paper/2507.06428/) (2025)
- [Convergence Guarantees for Neural Network-Based Hamilton-Jacobi Reachability](https://mlsanity.com/paper/2410.02904/) (2024)

## Abstract (arXiv)

We establish the convergence of the deep Galerkin method (DGM), a deep learning-based scheme for solving high-dimensional nonlinear PDEs, for Hamilton-Jacobi-Bellman (HJB) equations that arise from the study of mean field control problems (MFCPs). Based on a recent characterization of the value function of the MFCP as the unique viscosity solution of an HJB equation on the simplex, we establish both an existence and convergence result for the DGM. First, we show that the loss functional of the DGM can be made arbitrarily small given that the value function of the MFCP possesses sufficient regularity. Then, we show that if the loss functional of the DGM converges to zero, the corresponding neural network approximators must converge uniformly to the true value function on the simplex. We also provide numerical experiments demonstrating the DGM's ability to generalize to high-dimensional HJB equations.