The Fokker–Planck equation governs the uncertainty propagation of dynamical systems driven by stochastic processes. The solution of the Fokker–Planck equation is a time-varying Probability Density Function (PDF) that is usually of high dimension with unbounded support. There are also properties that must be conserved in time for the joint and any marginal solution PDF (i.e., a time-varying PDF is a non-negative function that must integrate to unity at any given time.) Satisfying these properties poses a significant challenge to the numerical solution of the Fokker–Planck equation. Another challenge is to capture the tail behavior of the solution PDF with unbounded support, required to predict the probability of rare events like a system failure. Satisfying the conditions of a proper PDF with unbounded support limits the application of traditional grid-based numerical methods, like finite difference and finite element methods. This paper develops a novel numerical method based on physics-based mixture models for the transient and steady-state solutions of the Fokker–Planck equation. In the proposed numerical method, the trial function space consists of the convex combinations of some parametric PDFs (i.e., mixture models) that trivially satisfy the necessary conditions of the solution. Estimating the unknown parameters of the mixture model is via Bayesian inference while considering the constraints on model parameters. Bayesian inference facilitates integrating data on the responses of dynamical systems (e.g., from simulations) with the governing Fokker–Planck equation to estimate the unknown parameters. Since the solution PDF is not observable, combining it with observable data on system responses is far from trivial based on current numerical methods. The formulation of Bayesian inference also introduces a weighting function to reduce the estimation error in specific regions of interest like the tail of the solution PDF. To reduce the computational demand, the paper develops an importance sampling algorithm that generates a small set of collocation points at which the residual of the Fokker–Planck equation is evaluated. The performance of the proposed numerical method is demonstrated with several benchmark problems. The obtained results are compared with analytical solutions when available and otherwise with simulations.

Fokker-Planck 方程控制由随机过程驱动的动力系统的不确定性传播。Fokker-Planck 方程的解是一个时变概率密度函数（PDF）通常具有无界支持的高维度。对于联合和任何边际解 PDF，还有一些属性必须及时保存（即，时变 PDF 是一个非负函数，必须在任何给定时间积分为单位。）满足这些属性提出了重大挑战Fokker-Planck 方程的数值解。另一个挑战是捕获具有无限支持的解决方案 PDF 的尾部行为，这是预测系统故障等罕见事件的概率所必需的。满足具有无限支持的适当 PDF 的条件限制了传统基于网格的数值方法的应用，例如有限差分和有限元方法。本文开发了一种基于物理混合模型的新型数值方法，用于 Fokker-Planck 方程的瞬态和稳态解。在所提出的数值方法中，试验函数空间由一些参数PDF 的凸组合（即混合模型）组成，它们通常满足解的必要条件。估计混合模型的未知参数是通过贝叶斯推理同时考虑模型参数的约束。贝叶斯推理有助于将有关动力系统响应的数据（例如，来自模拟）与控制 Fokker-Planck 方程相结合，以估计未知参数。由于解决方案 PDF 是不可观察的，因此基于当前的数值方法，将其与系统响应的可观察数据相结合绝非易事。贝叶斯推理的公式还引入了一个加权函数，以减少特定感兴趣区域（如解决方案 PDF 的尾部）的估计误差。为了减少计算需求，本文开发了一种重要性采样算法生成一小组配置点，在这些配置点处评估 Fokker-Planck 方程的残差。通过几个基准问题证明了所提出的数值方法的性能。获得的结果与可用的解析解进行比较，否则与模拟进行比较。