Solution of FPK Equation for Stochastic Dynamics Subjected to Additive Gaussian Noise via Deep Learning Approach
This addresses a challenging problem in engineering and physics for analyzing stochastic systems, offering a mesh-free, data-free approach that is incremental in applying deep learning to an existing numerical bottleneck.
The paper tackles solving the high-dimensional Fokker-Planck-Kolmogorov (FPK) equation for stochastic dynamics with additive Gaussian noise, introducing FPK-DP Net, a physics-informed deep learning method that achieves accurate results without prior simulation data, as demonstrated on five benchmark problems.
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.