In this paper we present an adaptive deep density approximation strategy based on KRnet (ADDA-KR) for solving the steady-state Fokker-Planck (F-P) equations. F-P equations are usually high-dimensional and defined on an unbounded domain, which limits 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 for the Fokker-Planck equations, which has a weaker dependence on dimensionality than traditional computational approaches and can efficiently estimate general high-dimensional density functions. To obtain effective stochastic collocation points for the approximation of the F-P equation, we develop an adaptive sampling procedure, where samples are generated iteratively using the approximate density function at each iteration. We present a general framework of ADDA-KR, validate its accuracy and demonstrate its efficiency with numerical experiments.

翻译：在本文中,我们提出了一个基于KRnet(ADA-KRR)的适应性深度密度近似战略,用于解决稳定状态Fokker-Planck(F-P)方程式。F-P方程式通常是高维的,定义在无约束域,限制了传统基于网格的数字方法的应用。Knothe-Rosenblatt的重新排列,我们新提出的流动基基因模型称为KRnet,提供了一种概率密度函数组合,作为Fokker-Planck方程式的有效解决方案候选方。Fokker-Planck方程式比传统的计算方法对维度的依赖弱,能够有效地估计一般高维密度功能。为了获得F-P方方方程式近似的有效蒸气共位,我们开发了一种适应性取样程序,即样品的迭接性生成在每次迭代中都使用大约的密度函数。我们提出了一个ADADA-KR的总框架,验证其准确性,并以数字实验来证明其效率。