Adaptive Sampling Methods for Physics-Informed Neural Networks
摘要
This study provides a comprehensive overview of key strategies for improving the performance of Physics-Informed Neural Networks (PINNs). We begin by outlining the general PINN framework, followed by a detailed presentation of three adaptive sampling techniques: Residual-based Adaptive Refinement (RAR), Monte Carlo Sampling for Failure Probability, and Self-Adaptive Importance Sampling (SAIS). To illustrate these methods in practice, we consider the one-dimensional heat equation as a benchmark problem. The associated partial differential equation is solved using both standard PINNs and SAIS-enhanced PINNs. A comparative analysis of the two approaches is presented in terms of training dynamics, including the number of epochs, convergence speed, and approximation accuracy. Then, we provide a brief overview of three other methods: Gradient-based methods, Residual-based Adaptive Distribution and Multi-criteria Adaptive Sampling. Finally, broader perspectives on performance enhancement strategies are discussed in the conclusion.