Connecting GANs, Mean-Field Games, and Optimal Transport

SIAM Journal on Applied Mathematics, Volume 84, Issue 4, Page 1255-1287, August 2024.
Abstract. Generative adversarial networks (GANs) have enjoyed tremendous success in image generation and processing and have recently attracted growing interest in financial modeling. This paper analyzes GANs from the perspectives of mean-field games (MFGs) and optimal transport: GANs are interpreted as MFGs under the Pareto optimality criterion or mean-field controls; meanwhile, GANs are to minimize the optimal transport cost indexed by the generator from the known latent distribution to the unknown true distribution of data. In particular, we provide a universal approximation result, which shows that there exists an appropriate neural network architecture for GANs training to capture the mean-field solution. The derivation of this universal approximation result leads to an explicit construction of the deep neural network for the transport mapping. The MFGs perspective of GANs leads to a GAN-based computational method (MFGANs) to solve MFGs: one neural network for the backward Hamilton–Jacobi–Bellman equation and one neural network for the forward Fokker–Planck equation, with the two neural networks trained in an adversarial way. Numerical experiments demonstrate superior performance of this proposed algorithm, especially in higher dimensional cases, when compared with existing neural network approaches.