An Adaptive Sampling Algorithm with Dynamic Iterative Probability Adjustment Incorporating Positional Information

Physics-informed neural networks (PINNs) have garnered widespread use for solving a variety of complex partial differential equations (PDEs). Nevertheless, when addressing certain specific problem types, traditional sampling algorithms still reveal deficiencies in efficiency and precision. In response, this paper builds upon the progress of adaptive sampling techniques, addressing the inadequacy of existing algorithms to fully leverage the spatial location information of sample points, and introduces an innovative adaptive sampling method. This approach incorporates the Dual Inverse Distance Weighting (DIDW) algorithm, embedding the spatial characteristics of sampling points within the probability sampling process. Furthermore, it introduces reward factors derived from reinforcement learning principles to dynamically refine the probability sampling formula. This strategy more effectively captures the essential characteristics of PDEs with each iteration. We utilize sparsely connected networks and have adjusted the sampling process, which has proven to effectively reduce the training time. In numerical experiments on fluid mechanics problems, such as the two-dimensional Burgers' equation with sharp solutions, pipe flow, flow around a circular cylinder, lid-driven cavity flow, and Kovasznay flow, our proposed adaptive sampling algorithm markedly enhances accuracy over conventional PINN methods, validating the algorithm's efficacy.