Practical Aspects of Physics-Informed Neural Networks Applied to Solve Frequency-Domain Acoustic Wave Forward Problem

Physics-informed neural networks (PINNs) have been used by researchers to solve partial differential equation (PDE)-constrained problems. We evaluate PINNs to solve for frequency-domain acoustic wavefields. PINNs can solely use PDEs to define the loss function for optimization without the need for labels. Partial derivatives of PDEs are calculated by mesh-free automatic differentiations. Thus, PINNs are free of numerical dispersion artifacts. It has been applied to the scattered acoustic wave equation, which relied on boundary conditions (BCs) provided by the background analytical wavefield. For a more direct implementation, we solve the nonscattered acoustic wave equation, avoiding limitations related to relying on the background homogeneous medium for BCs. Experiments support our following insights. Although solving time-domain wave equations using PINNs does not require absorbing boundary conditions (ABCs), ABCs are required to ensure a unique solution for PINNs that solve frequency-domain wave equations, because the single-frequency wavefield is not localized and contains wavefield information over the full domain. However, it is not trivial to include the ABC in the PINN implementation, so we develop an adaptive amplitude-scaled and phase-shifted sine activation function, which performs better than the previous implementations. Because there are only two outputs for the fully connected neural network (FCNN), we validate a linearly shrinking FCNN that can achieve a comparable and even better accuracy with a cheaper computational cost. However, there is a spectral bias problem, that is, PINNs learn low-frequency wavefields far more easily than higher frequencies, and the accuracy of higher frequency wavefields is often poor. Because the shapes of multifrequency wavefields are similar, we initialize the FCNN for higher frequency wavefields by that of the lower frequencies, partly mitigating the spectral bias problem. We further incorporate multiscale positional encoding to alleviate the spectral bias problem. We share our codes, data, and results via a public repository.