Physics-Enhanced Deep Fourier Network for Seismic Wave Forward Modelling

Forward modelling of seismic wavefields is a cornerstone of geophysical studies, aiding in subsurface characterization and exploration. In this study, we introduce a Physics-Enhanced Deep Fourier-Attention Network (PE-DFAN) to simulate the forward process from physical property differences to wavefields, addressing the limitations of conventional neural networks in capturing complex wavefield patterns. Conventional neural networks often struggle to model the intricate spatial correlations inherent in wave propagation, resulting in a tendency to learn only the average field behaviour. To overcome this, we incorporate a Fourier attention layer that learns coordinate correlations effectively and expands input coordinates into a high-dimensional Fourier space. This design enhances the network's ability to represent fine-grained spatial variations. Furthermore, our model outperforms both standard neural networks and purely Fourier-feature-based networks in predictive accuracy. To ensure higher-order physical consistency, we introduce a frequency-domain-based acoustic equation as an additional constraint in the loss function. This physics-informed approach enforces adherence to acoustic equation principles, leading to improved alignment with theoretical expectations. Experimental results demonstrate that the PE-DFAN achieves superior performance in both accuracy and physical fidelity, marking a significant advancement in neural network-based seismic forward modelling. This work underscores the potential of combining advanced neural network architectures with physics-based constraints, paving the way for more precise and computationally efficient seismic modelling frameworks.