# Deep Learning for Mean Field Games with non-separable Hamiltonians

arXiv: https://arxiv.org/abs/2301.02877 · HTML: https://mlsanity.com/paper/2301.02877/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 47 references that are in the map.
- First posted on arXiv 7 Jan 2023.

## Builds on (most cited first, 16 of 16)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Generative Adversarial Text to Image Synthesis](https://mlsanity.com/paper/1605.05396/) (2016)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Deep Generative Image Models using a Laplacian Pyramid of Adversarial Networks](https://mlsanity.com/paper/1506.05751/) (2015)
- [A Mean-Field Optimal Control Formulation of Deep Learning](https://mlsanity.com/paper/1807.01083/) (2018)
- [Deep Hedging: Learning to Simulate Equity Option Markets](https://mlsanity.com/paper/1911.01700/) (2019)
- [Mean Field Game of Controls and An Application To Trade Crowding](https://mlsanity.com/paper/1610.09904/) (2016)
- [Deep Learning for Mean Field Games and Mean Field Control with Applications to Finance](https://mlsanity.com/paper/2107.04568/) (2021)
- [Massive Autonomous UAV Path Planning: A Neural Network Based Mean-Field Game Theoretic Approach](https://mlsanity.com/paper/1905.04152/) (2019)
- [A Game-Theoretic Framework for Autonomous Vehicles Velocity Control: Bridging Microscopic Differential Games and Macroscopic Mean Field Games](https://mlsanity.com/paper/1903.06053/) (2019)
- [Policy iteration method for time-dependent Mean Field Games systems with non-separable Hamiltonians](https://mlsanity.com/paper/2110.02552/) (2021)
- [An integral control formulation of Mean-field game based large scale coordination of loads in smart grids](https://mlsanity.com/paper/1803.00040/) (2018)
- [Algorithm for Hamilton-Jacobi equations in density space via a generalized Hopf formula](https://mlsanity.com/paper/1805.01636/) (2018)

## Built on by (most cited first, 1 of 1)

- [Neural Mean-Field Games: Extending Mean-Field Game Theory with Neural Stochastic Differential Equations](https://mlsanity.com/paper/2504.13228/) (2025)

## Abstract (arXiv)

This paper introduces a new method based on Deep Galerkin Methods (DGMs) for solving high-dimensional stochastic Mean Field Games (MFGs). We achieve this by using two neural networks to approximate the unknown solutions of the MFG system and forward-backward conditions. Our method is efficient, even with a small number of iterations, and is capable of handling up to 300 dimensions with a single layer, which makes it faster than other approaches. In contrast, methods based on Generative Adversarial Networks (GANs) cannot solve MFGs with non-separable Hamiltonians. We demonstrate the effectiveness of our approach by applying it to a traffic flow problem, which was previously solved using the Newton iteration method only in the deterministic case. We compare the results of our method to analytical solutions and previous approaches, showing its efficiency. We also prove the convergence of our neural network approximation with a single hidden layer using the universal approximation theorem.