Solution of FPK Equation for Stochastic Dynamics Subjected to Additive
  Gaussian Noise via Deep Learning Approach
Amir H. Khodabakhsh Seid H. Pourtakdoust
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.