# Deep Backward and Galerkin Methods for the Finite State Master Equation

arXiv: https://arxiv.org/abs/2403.04975 · HTML: https://mlsanity.com/paper/2403.04975/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 48 references that are in the map.
- First posted on arXiv 8 Mar 2024.

## Builds on (most cited first, 24 of 24)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020)
- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [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)
- [Generalization in Mean Field Games by Learning Master Policies](https://mlsanity.com/paper/2109.09717/) (2021)
- [Deep Learning Methods for Mean Field Control Problems with Delay](https://mlsanity.com/paper/1905.00358/) (2019)
- [Recent Developments in Machine Learning Methods for Stochastic Control and Games](https://mlsanity.com/paper/2303.10257/) (2023)
- [Neural networks-based backward scheme for fully nonlinear PDEs](https://mlsanity.com/paper/1908.00412/) (2019)
- [Mean Field Games and Applications: Numerical Aspects](https://mlsanity.com/paper/2003.04444/) (2020)
- [Numerical Methods for Mean Field Games and Mean Field Type Control](https://mlsanity.com/paper/2106.06231/) (2021)
- [Numerical resolution of McKean-Vlasov FBSDEs using neural networks](https://mlsanity.com/paper/1909.12678/) (2019)
- [Extensions of the Deep Galerkin Method](https://mlsanity.com/paper/1912.01455/) (2019)
- [DeepSets and their derivative networks for solving symmetric PDEs](https://mlsanity.com/paper/2103.00838/) (2021)
- [Mean-field-game model for Botnet defense in Cyber-security](https://mlsanity.com/paper/1511.06642/) (2015)
- [Connecting GANs, MFGs, and OT](https://mlsanity.com/paper/2002.04112/) (2020)
- [A numerical scheme for a mean field game in some queueing systems based on Markov chain approximation method](https://mlsanity.com/paper/1708.08343/) (2017)
- [Rate of convergence for particle approximation of PDEs in Wasserstein space](https://mlsanity.com/paper/2103.00837/) (2021)
- [Analysis of a Finite State Many Player Game Using its Master Equation](https://mlsanity.com/paper/1707.02648/) (2017)
- [Mean field games master equations: from discrete to continuous state space](https://mlsanity.com/paper/2207.03191/) (2022)
- [Analysis of the Finite-State Ergodic Master Equation](https://mlsanity.com/paper/2204.05373/) (2022)

## Built on by (most cited first, 6 of 6)

- [Convergence of the Deep Galerkin Method for Finite State Mean Field Control Problems](https://mlsanity.com/paper/2405.13346/) (2024)
- [Global Solutions to Master Equations for Continuous Time Heterogeneous Agent Macroeconomic Models](https://mlsanity.com/paper/2406.13726/) (2024)
- [Operator Learning for Families of Finite-State Mean-Field Games](https://mlsanity.com/paper/2602.13169/) (2026)
- [Convergence Guarantees for Neural Network-Based Hamilton-Jacobi Reachability](https://mlsanity.com/paper/2410.02904/) (2024)
- [Neural Parameter Calibration for Finite-State Mean Field Games](https://mlsanity.com/paper/2606.23155/) (2026)
- [Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs](https://mlsanity.com/paper/2607.24726/) (2026)

## Abstract (arXiv)

This paper proposes and analyzes two neural network methods to solve the master equation for finite-state mean field games (MFGs). Solving MFGs provides approximate Nash equilibria for stochastic, differential games with finite but large populations of agents. The master equation is a partial differential equation (PDE) whose solution characterizes MFG equilibria for any possible initial distribution. The first method we propose relies on backward induction in a time component while the second method directly tackles the PDE without discretizing time. For both approaches, we prove two types of results: there exist neural networks that make the algorithms' loss functions arbitrarily small, and conversely, if the losses are small, then the neural networks are good approximations of the master equation's solution. We conclude the paper with numerical experiments on benchmark problems from the literature up to dimension 15, and a comparison with solutions computed by a classical method for fixed initial distributions.