Adaptive deep density approximation for Fokker-Planck equations
Adaptive deep density approximation for Fokker-Planck equations
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DOI：
10.1016/j.jcp.2022.111080
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发表时间：
2021-03
期刊：
影响因子：
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通讯作者：
Keju Tang;X. Wan;Qifeng Liao
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文献类型：
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作者：
Keju Tang;X. Wan;Qifeng Liao
在本文中，我们提出了一种基于 KRnet (ADDA-KR) 的自适应深度密度逼近策略，用于求解稳态 Fokker-Planck (F-P) 方程。 F-P方程通常是高维的并且在无界域上定义，这限制了传统基于网格的数值方法的应用。通过 Knothe-Rosenblatt 重排，我们新提出的基于流的生成模型（称为 KRnet）提供了一系列概率密度函数，可作为 Fokker-Planck 方程的有效候选解，与传统计算方法相比，它对维数的依赖性较弱，并且可以有效地估计一般的高维密度函数。为了获得 F-P 方程近似的有效随机配置点，我们开发了一种自适应采样程序，其中在每次迭代时使用近似密度函数迭代生成样本。我们提出了 ADDA-KR 的总体框架，验证了其准确性并通过数值实验证明了其效率。
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.