A Cross-Gating PINN with Physics-Guided Multi-Scale Spatial Phase for Wavefield Simulation

Physics-Informed Neural Networks (PINNs), as mesh-free solvers for partial differential equations, often encounter the high-frequency fitting predicament caused by spectral bias and local wavefront distortion when solving frequency-domain wave equations (Helmholtz equations). Moreover, traditional data-driven random weight initialization easily introduces unphysical artifacts and computational redundancy in feature space. To address these issues, this paper proposes a novel neural wavefield solver, CG-MSP-PINN, based on physical priors of multi-scale spatial phase and cross-gating mechanisms. First, the physical dispersion relation of the wave equation is explicitly mapped onto Cartesian wave vectors, constructing a feature distribution in wavenumber space with strict physical boundaries to suppress non-physical noise. Second, a dual-band physical feature dictionary is designed: global broadband absolute phase is used to fit macroscopic smooth background, while local single-frequency relative phase captures abrupt variations in the target dominant frequency band. Furthermore, a dual-branch cross-gating fusion architecture is introduced, enabling synergistic representation through bidirectional nonlinear modulation between implicit wavefield characterization via sinusoidal representation networks (SIREN) and explicit multi-scale physical dictionaries. Numerical experiments across frequency bands using Simple and Marmousi models demonstrate that CG-MSP-PINN effectively mitigates scale aliasing problems. Compared to pure SIREN architectures, this method achieves higher wavefield reconstruction accuracy while reducing dependence on spatial collocation density and training duration. This study provides a deep learning solution that takes into account both physical constraints and computational efficiency for seismic exploration and full waveform inversion.