Simulating seismic multifrequency wavefields with the Fourier feature physics-informed neural network

To simulate seismic wavefields with a frequency-domain wave equation, conventional numerical methods must solve the equation sequentially to obtain the wavefields for different frequencies. The monofrequency equation has the form of a Helmholtz equation. When solving the Helmholtz equation for seismic wavefields with multiple frequencies, a physics-informed neural network (PINN) can be used. However, the PINN suffers from the problem of spectral bias when approximating high-frequency components. We propose to simulate seismic multifrequency wavefields using a PINN with an embedded Fourier feature. The input to the Fourier feature PINN for simulating multifrequency wavefields is 4-D, namely the horizontal and vertical spatial coordinates of the model, the horizontal position of the source, and the frequency, and the output is multifrequency wavefields at arbitrary source positions. While an effective Fourier feature initialization strategy can lead to optimal convergence in training this network, the Fourier feature PINN simulates multifrequency wavefields with reasonable efficiency and accuracy.