Abstract: Aiming at the problem of insufficient accuracy of traditional Physical information neural networks (PINNs) when dealing with high-frequency features, this paper proposes a two-stage PINNs method based on adaptive weights (AWTS-PINNs) to solve partial differential equations with high-frequency solutions. This method is based on a two-stage training framework combining pre-training and fine-tuning. It introduces an activation function with the response ability of high-frequency features and integrates the adaptive mechanism of neural tangent kernels to dynamically adjust the weight of the loss function, thereby significantly improving the model’s ability to express and capture high-frequency features. The experimental results show that this method performs well in capturing high-frequency features. Compared with the existing methods such as PINNs, NTK-PINNS, RFF-PINNS and DG-PINNs, AWTS-PINNs has higher accuracy and solution efficiency. In one-dimensional and two-dimensional numerical experiments, AWTS-PINNs achieves the lowest test errors, with an accuracy reaching the order of 10^-4.

Key words: PINNs, high-frequency solution, partial differential equations (PDEs), spectral bias, neural tangent kernel