PINN-based Seismic Wavefield Simulation for the Helmholtz Equation Parameterized by Amplitude and Unwrapped Phase

Abstract Seismic wavefield simulation serves as an essential step for high-resolution seismic imaging. Traditional methods like the finite difference method (FDM) require dense grid meshing for complex velocity models. In 3D frequency-domain wavefield simulation, FDM needs to invert a large impedance matrix. This process is extremely slow and consumes substantial computational resources. As an emerging mesh-free paradigm for forward modeling, physics-informed neural networks (PINNs) adopt physical principles as constraints to optimize the network and have demonstrated significant potential in simulating seismic wavefields for complex models. However, hindered by the inherent spectral bias of neural networks, PINNs struggle to capture the high-frequency spatial oscillations of complex-valued seismic wavefields for the Helmholtz equation, leading to decreased accuracy and slow convergence. To overcome this challenge, we propose to solve a new form of the Helmholtz equation parameterized by amplitude and unwrapped phase with PINNs. Consequently, the frequency-domain wavefield will be represented by amplitude and unwrapped phase, which is significantly smoother, resulting in a significantly improved convergence in training PINNs. Numerical experiments on 2D and 3D complex models demonstrate that, compared with the conventional Helmholtz equations based on complex-valued wavefield representation, the proposed method significantly enhances convergence speed and wavefield simulation accuracy, even when utilizing simple fully connected networks.