The Fokker-Plank-Kolmogorov (FPK) equation is an idealized model representing many stochastic systems commonly encountered in the analysis of stochastic structures as well as many other applications. Its solution thus provides an invaluable insight into the performance of many engineering systems. Despite its great importance, the solution of the FPK equation is still extremely challenging. For systems of practical significance, the FPK equation is usually high dimensional, rendering most of the numerical methods ineffective. In this respect, the present work introduces the FPK-DP Net as a physics-informed network that encodes the physical insights, i.e. the governing constrained differential equations emanated out of physical laws, into a deep neural network. FPK-DP Net is a mesh-free learning method that can solve the density evolution of stochastic dynamics subjected to additive white Gaussian noise without any prior simulation data and can be used as an efficient surrogate model afterward. FPK-DP Net uses the dimension-reduced FPK equation. Therefore, it can be used to address high-dimensional practical problems as well. To demonstrate the potential applicability of the proposed framework, and to study its accuracy and efficacy, numerical implementations on five different benchmark problems are investigated.

翻译：Fokker-Plank-Kolmogorov（FPK）方程是描述随机结构分析及其他众多应用中常见随机系统的理想化模型，其解能为众多工程系统的性能评估提供宝贵见解。尽管具有重大意义，FPK方程的求解仍极具挑战性。对于具有实际重要性的系统，FPK方程通常具有高维特性，这导致多数数值方法失效。为此，本文提出FPK-DP Net作为物理信息网络，将物理规律导出的约束微分方程等物理认知编码至深度神经网络中。FPK-DP Net是一种无网格学习方法，无需任何先验仿真数据即可求解加性高斯白噪声下随机动态系统的密度演化过程，并能作为高效替代模型。通过采用降维FPK方程，该方法亦可处理高维实际问题。为验证所提框架的适用性并评估其精度和有效性，本文对五个基准问题进行了数值实现研究。