Solving Wave Equations with Physics-Informed Neural Networks Integrating Fourier Features and Adaptive Weights

To mitigate the challenge that conventional physics-informed neural networks (PINN) face in effectively capture high-frequency information when solving wave equations, resulting in slow convergence and limited accuracy, an improved PINN model, termed adaptive weight and Fourier feature PINN (AWF-PINN), was proposed. In this approach, a learnable Fourier feature mapping was incorporated into the input layer to enhance the model’s capability for high-frequency oscillatory features. Additionally, an adaptive weight strategy based on exponential normalization of loss change rates was designed to dynamically balance multiple losses, including the PDE residual, initial conditions, and boundary conditions, thereby improving optimization stability. To validate the performance of the proposed model, numerical experiments were conducted on the wave equation and the Klein-Gordon equation as test cases.The results demonstrate that, compared with the traditional PINN, AWF-PINN achieves significant performance improvements on both equations, with substantially fewer iterations required for convergence and a reduction in relative errors by over 80%, thereby effectively mitigating the issues of spectral bias and loss term imbalance. By integrating Fourier feature embedding with weight adaptation, this study provides an efficient and robust approach for solving partial differential equations exhibiting periodic and high-frequency characteristics, thus broadening the applicability of PINNs to complex wave problems.