DSA-PINN: A dynamic sample-adapted physics-informed neural network for nonlinear Fokker–Planck equations

The Fokker–Planck (FP) equation is a fundamental component in the mathematical modeling of stochastic processes which governs the time development of probability density functions in a variety of fields, including psychology, neuroscience, statistical physics, quantitative finance, and systems biology. Despite its significance, finding exact solutions to nonlinear FP equations with time and space dependent coefficients is still very difficult, especially in high dimensions when using classical discretization methods becomes computationally expensive. In this work, we introduce an enhanced Physics-Informed Neural Network (PINN) framework with added Dynamic Sample Adaptation (DSA) to address these difficulties. The proposed method by adaptively moving collocation points during training, concentrating computational effort in point where the solution has sharp gradients, high variability, or localized peaks. Unlike standard PINNs, where collocation points remain fixed throughout training, the proposed DSA strategy dynamically relocates collocation points during the optimization process based on the PDE residual. This adaptive sampling technique works by updating the sampling points using either the Adam optimizer or gradient descent after a few training epochs to ensure both global exploration and fine-scale accuracy. By integrating physics-informed loss functions with dynamically refined training samples, the framework achieves improved stability and convergence when compared to standard PINNs. An example set of seven problems is used to thoroughly evaluate the effectiveness of the proposed approach. This set includes two and three dimensional challenging nonlinear FP equations with stiff initial conditions and time varying variables. A comparison with standard PINN and the traditional numerical approach is carried out. The DSA-PINN method exhibits better accuracy, quicker convergence, and increased robustness in capturing intricate solution structures in most test cases.