Adaptive sampling points based multi-scale residual network for solving partial differential equations

Physics-informed neural networks (PINNs) have shown remarkable achievements in solving partial differential equations (PDEs). However, their performance is limited when encountering oscillatory part in the solutions of PDEs. Therefore, this paper proposes a multi-scale deep neural network with periodic activation function to achieve high-frequency to low-frequency conversion, which can capture the oscillation part of the solution of PDEs. Moreover, the use of adaptive sampling method can adaptively change the location and distribution of residual points, improving the performance of the network. Additionally, the gradient-enhanced strategy is also utilized to embed the gradient information of the PDEs into the loss function of the neural network, which further improves the accuracy of PINNs. Through the numerical experiments verification, it is found that our method is better than PINNs in terms of accuracy and efficiency.