Connecting GANs, MFGs, and OT
This work provides a theoretical framework linking GANs to MFGs and optimal transport, offering a novel method for solving MFGs, which is incremental but domain-specific to computational mathematics and finance.
This paper analyzes Generative Adversarial Networks (GANs) by connecting them to mean-field games (MFGs) and optimal transport, leading to a new computational method called MFGANs for solving MFGs. Numerical experiments show that MFGANs outperform existing neural network approaches, particularly in higher-dimensional cases.
Generative adversarial networks (GANs) have enjoyed tremendous success in image generation and processing, and have recently attracted growing interests in financial modelings. This paper analyzes GANs from the perspectives of mean-field games (MFGs) and optimal transport. More specifically, from the game theoretical perspective, GANs are interpreted as MFGs under Pareto Optimality criterion or mean-field controls; from the optimal transport perspective, GANs are to minimize the optimal transport cost indexed by the generator from the known latent distribution to the unknown true distribution of data. 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 the higher dimensional case, when compared with existing neural network approaches.