Adaptive deep density approximation for Fokker-Planck equations

In this paper we present a novel adaptive deep density approximation strategy based on KRnet (ADDA-KR) for solving the steady-state Fokker-Planck equation. It is known that this equation typically has high-dimensional spatial variables posed on unbounded domains, which limit the application of traditional grid based numerical methods. With the Knothe-Rosenblatt rearrangement, our newly proposed flow-based generative model, called KRnet, provides a family of probability density functions to serve as effective solution candidates of the Fokker-Planck equation, which have weaker dependence on dimensionality than traditional computational approaches. To result in effective stochastic collocation points for training KRnet, we develop an adaptive sampling procedure, where samples are generated iteratively using KRnet at each iteration. In addition, we give a detailed discussion of KRnet and show that it can efficiently estimate general high-dimensional density functions. We present a general mathematical framework of ADDA-KR, validate its accuracy and demonstrate its efficiency with numerical experiments.