A spatial–temporal normalization framework for Fourier-enhanced physics-informed deep learning in large-scale wave propagation

Abstract Physics-informed neural networks (PINNs) have emerged as promising artificial intelligence-driven approaches for solving wave propagation equations in scientific and engineering applications such as seismic simulation, fluid dynamics, and electromagnetics. However, conventional PINNs frameworks often suffer from optimization instability, spectral bias, and degraded predictive accuracy when solving wave equations in large-scale spatial domains and over extended temporal ranges. These limitations mainly arise from scale disparity in spatial–temporal coordinates and the difficulty of learning multi-scale wave features using fully connected neural networks. To address these challenges, this study proposes a spatial-temporal normalization framework for Fourier-enhanced physics-informed deep learning in large-scale wave propagation problems. The proposed framework integrates spatial and temporal normalization strategies into a Fourier-enhanced PINNs architecture to improve optimization conditioning, training stability, and the representation of high-frequency features. Fourier feature mapping (FFM) is incorporated to alleviate spectral bias and strengthen the learning capability of oscillatory wave behaviors in complex spatial–temporal domains. Different normalization strategies are systematically investigated to identify the most effective learning configuration for large-scale wave simulations. The effectiveness and robustness of the proposed framework are validated through five representative numerical experiments in both two-dimensional and three-dimensional regular and irregular domains. Experimental results demonstrate that the proposed method achieves significantly improved convergence behavior, predictive accuracy, and stability compared with conventional PINNs-based approaches. The proposed framework provides an effective artificial intelligence-driven computational approach for large-scale wave propagation modeling and shows strong potential for broader engineering applications.