Physics-informed neural network for random response evaluation

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 3dimensional 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-PlanckKolmogorov (FPK) equation with high precision. We believe this method can effectively address the FPK equation for various random dynamical systems.