Deep Learning for Mean Field Games with non-separable Hamiltonians

TL;DR

This paper presents a deep learning approach using Deep Galerkin Methods to efficiently solve high-dimensional stochastic Mean Field Games with non-separable Hamiltonians, outperforming existing methods in speed and capability.

Contribution

Introduces a neural network-based method for high-dimensional MFGs with non-separable Hamiltonians, demonstrating efficiency and convergence proof.

Findings

Capable of solving up to 300 dimensions with a single layer neural network.

Outperforms GAN-based methods in handling non-separable Hamiltonians.

Validated on a traffic flow problem with results matching analytical solutions.

Abstract

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.