Learning Neural Pushforward Samplers for Distributions from Fokker-Planck Equations by Weak Adversarial Training

This paper presents a new method for solving Fokker-Planck equations (FPE) by learning a neural sampler for the distribution given by the FPE via an adversarial training based on a weak formulation of the FPE where the adjoint operator of FPE acts on the test function. Such a weak formulation transforms the PDE solution problem into a Monte Carlo importance sampling problem where the FPE solution-distribution is learned through a neural pushforward map, avoiding some of the limitations of direct PDE based methods. Moreover, by using simple plane-wave test functions, derivatives on the test functions can be explicitly computed. This approach produces a natural importance sampling strategy for the FPE solution distribution with probability conservation, from which the FPE solution can be easily constructed.