mlsanityData to 7 Oct 2026

Deep Backward and Galerkin Methods for the Finite State Master Equation

arXiv 2403.04975 ↗First posted 8 Mar 2024math.OC stat.MLcited by 6

Not on the map. Why?

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.

Key facts. Cited by 6 papers in the map (data through 7 Oct 2026). Builds on 24 of its 48 references that are in the map. First posted on arXiv 8 Mar 2024.

Builds on

24 of 48 references.

References · 24

  1. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  2. 2018Regularisation of Neural Networks by Enforcing Lipschitz Continuitycited by 311
  3. 2019Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networkscited by 290
  4. 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 188
  5. 2020Training robust neural networks using Lipschitz boundscited by 98
  6. 2019A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problemscited by 95
  7. 2019Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: II -- The Finite Horizon Casecited by 34
  8. 2016Minimizing the Maximal Loss: How and Why?cited by 30
  9. 2021Generalization in Mean Field Games by Learning Master Policiescited by 28
  10. 2019Deep Learning Methods for Mean Field Control Problems with Delaycited by 24
  11. 2020Mean Field Games and Applications: Numerical Aspectscited by 22
  12. 2019Neural networks-based backward scheme for fully nonlinear PDEscited by 21
12 more
  1. 2021Numerical Methods for Mean Field Games and Mean Field Type Controlcited by 21
  2. 2019Numerical resolution of McKean-Vlasov FBSDEs using neural networkscited by 19
  3. 2023Recent Developments in Machine Learning Methods for Stochastic Control and Gamescited by 19
  4. 2019Extensions of the Deep Galerkin Methodcited by 10
  5. 2021DeepSets and their derivative networks for solving symmetric PDEscited by 10
  6. 2020Connecting GANs, MFGs, and OTcited by 8
  7. 2015Mean-field-game model for Botnet defense in Cyber-securitycited by 7
  8. 2017A numerical scheme for a mean field game in some queueing systems based on Markov chain approximation methodcited by 3
  9. 2021Rate of convergence for particle approximation of PDEs in Wasserstein spacecited by 3
  10. 2017Analysis of a Finite State Many Player Game Using its Master Equationcited by 2
  11. 2022Mean field games master equations: from discrete to continuous state spacecited by 2
  12. 2022Analysis of the Finite-State Ergodic Master Equationcited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 3; 2025: 0; 2026: 3.

6 papers.

  1. 2024Convergence of the Deep Galerkin Method for Finite State Mean Field Control Problemscited by 2
  2. 2024Global Solutions to Master Equations for Continuous Time Heterogeneous Agent Macroeconomic Modelscited by 2
  3. 2026Operator Learning for Families of Finite-State Mean-Field Gamescited by 2
  4. 2024Convergence Guarantees for Neural Network-Based Hamilton-Jacobi Reachabilitycited by 0
  5. 2026Neural Parameter Calibration for Finite-State Mean Field Gamescited by 0
  6. 2026Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEscited by 0