Physics-informed neural network for random response evaluation

Abstract

In this paper, we propose a physics-informed neural network algorithm (PINN) to solve Fokker–Planck–Kolmogorov (FPK) equations for stochastic dynamical systems. The primary innovation of our approach lies in decomposing the solution of the FPK equations into two components: the probability density function (PDF) of the associated degenerate systems, derived from prior knowledge, and a modified component expressed in exponential form. This decomposition provides several advantages. First, the normalization condition as a supervisory criterion to prevent a zero solution is unnecessary, which reduces computational costs during the gradient descent iteration process, particularly in high-dimensional systems. Second, this approach accommodates uneven sample points. Third, the boundary condition is automatically satisfied. We present numerical examples to demonstrate the effectiveness of the proposed physics-informed neural networks. By utilizing 2- or 3-dimensional systems as examples, comparisons with exact solutions and results from Monte Carlo simulations show strong agreement, indicating that the physics-informed neural networks can solve the Fokker-Planck-Kolmogorov (FPK) equation with high precision. We believe this method can effectively address the FPK equation for various random dynamical systems.