  We propose a novel mean field games (MFGs) based GAN(generative adversarial
network) framework. To be specific, we utilize the Hopf formula in density
space to rewrite MFGs as a primal-dual problem so that we are able to train the
model via neural networks and samples. Our model is flexible due to the freedom
of choosing various functionals within the Hopf formula. Moreover, our
formulation mathematically avoids Lipschitz-1 constraint. The correctness and
efficiency of our method are validated through several experiments.
