AN ADAPTIVE COLLOCATION POINT ALGORITHM FOR PHYSICS-INFORMED NEURAL NETWORKS

Physics-informed neural networks (PINNs) integrate equation models into loss minimization training, enabling simultaneous learning of input data distributions and physical laws. Most PINNs employ a uniform sampling strategy to cover the entire solution domain, with each collocation point playing an equal role in the training process. While this straightforward configuration point strategy is easy to implement, it often leads PINNs to incorporate unnecessary collocation points and demonstrates insufficient learning capability for certain complex laws. In this paper, we propose an adaptive collocation point selection strategy to enhance the learning capability and efficiency of PINNs. Firstly, the selection probability of collocation points is determined through the joint distribution of the residuals of the loss function and their gradients. Additionally, after a certain number of iterations, resampling is conducted to avoid premature convergence to local optima. This approach allows some collocation points to be distributed in regions with high loss or significant variation, thereby improving the overall distribution of collocation points. As a result, the model can accurately reflect the equation with fewer collocation points, thus enhancing learning efficiency. Secondly, we introduce a variable weight setting for collocation points, allowing each point to exert varying degrees of influence on the equation's residuals. During the network's iterative training, the weights of collocation points associated with higher loss values are automatically increased, enabling the PINN to concentrate more on the aspects with greater loss—namely, the learning of complex laws. Finally, comparative experiments are conducted using four typical cases: the Burgers' equation, the Schrödinger equation, the Helmholtz equation, and the Navier-Stokes equations, benchmarked against traditional PINNs and various improved methods. Numerical results demonstrate that the proposed algorithm effectively enhances solution accuracy and computational efficiency under conditions of fewer collocation points and iterations.