Physics-Informed Neural Networks with hybrid sampling for stationary Fokker–Planck–Kolmogorov Equation

The Fokker–Planck–Kolmogorov (FPK) equation provides a deterministic framework for describing the evolution of probability density functions (PDF) and holds great significance in the field of stochastic dynamics. Physics-Informed Neural Networks (PINNs) development provides a new approach for solving the FPK equation. However, the vanilla PINNs method with uniform sampling faces challenges when dealing with FPK equations with strong nonlinear terms whose solutions have high frequency. In this study, we introduce PINNs with hybrid sampling specifically designed to solve the stationary FPK equation. The proposed hybrid sampling specifically introduces trajectory residual points and combines them with an adaptive sampling of residual points based on the residuals of the equation. In addition, the network incorporates a customized architecture and loss function tailored to address the challenges and specific requirements associated with solving the stationary FPK equation. We apply the proposed hybrid sampling method to three nonlinear systems, demonstrating its accuracy and efficiency by comparing it with Latin hypercube sampling (LHS), adaptive sampling, and trajectory sampling. The impact of the nonlinear parameters on the performance of different sampling methods is analyzed, emphasizing the superior accuracy achieved by hybrid sampling in cases involving significant nonlinearity. These experimental results demonstrate that incorporating trajectory residual points notably enhances computational performance and precision. Furthermore, we analyze the impact of different trajectory points and adaptive points on network training and examine the effect of the learning rate strategy on the performance of the proposed method.