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.

Fokker-Plank-Kolmogorov (FPK) 方程是一个理想化模型，代表随机结构分析以及许多其他应用中常见的许多随机系统。因此，它的解决方案为许多工程系统的性能提供了宝贵的见解。尽管 FPK 方程非常重要，但其求解仍然极具挑战性。对于具有实际意义的系统，FPK方程通常是高维的，使得大多数数值方法无效。在这方面，目前的工作引入了 FPK-DP Net 作为物理信息网络，它将物理见解（即从物理定律产生的控制约束微分方程）编码到深度神经网络中。FPK-DP Net是一种无网格学习方法，可以在没有任何事先模拟数据的情况下解决加性高斯白噪声下随机动力学的密度演化，并且可以在事后用作有效的代理模型。FPK-DP Net 使用降维 FPK 方程。因此，它也可以用来解决高维的实际问题。为了证明所提出的框架的潜在适用性，并研究其准确性和有效性，研究了五个不同基准问题的数值实现。