Neural Parametric Fokker-Planck Equation

Shu Liu, Wuchen Li, Hongyuan Zha, Haomin Zhou

Abstract: 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.