  In this paper, we develop and analyze numerical methods for high dimensional
Fokker-Planck equations by leveraging generative models from deep learning. Our
starting point is a formulation of the Fokker-Planck equation as a system of
ordinary differential equations (ODEs) on finite-dimensional parameter space
with the parameters inherited from generative models such as normalizing flows.
We call such ODEs neural parametric Fokker-Planck equations. The fact that the
Fokker-Planck equation can be viewed as the $L^2$-Wasserstein gradient flow of
Kullback-Leibler (KL) divergence allows us to derive the ODEs as the
constrained $L^2$-Wasserstein gradient flow of KL divergence on the set of
probability densities generated by neural networks. For numerical computation,
we design a variational semi-implicit scheme for the time discretization of the
proposed ODE. Such an algorithm is sampling-based, which can readily handle the
Fokker-Planck equations in higher dimensional spaces. Moreover, we also
establish bounds for the asymptotic convergence analysis of the neural
parametric Fokker-Planck equation as well as the error analysis for both the
continuous and discrete versions. Several numerical examples are provided to
illustrate the performance of the proposed algorithms and analysis.
