A versatile framework to solve the Helmholtz equation using physics-informed neural networks

Solving the wave equation to obtain wavefield solutions is an essential step in illuminating the subsurface using seismic imaging and waveform inversion methods. Here, we utilize a recently introduced machine-learning based framework called physics-informed neural networks (PINNs) to solve the frequency-domain wave equation, which is also referred to as the Helmholtz equation, for isotropic and anisotropic media. Like functions, PINNs are formed by using a fully-connected neural network (NN) to provide the wavefield solution at spatial points in the domain of interest, in which the coordinates of the point form the input to the network. We train such a network by back propagating the misfit in the wave equation for the output wavefield values and their derivatives for many points in the model space. Generally, a hyperbolic tangent activation is used with PINNs, however, we use an adaptive sinusoidal activation function to optimize the training process. Numerical results show that PINNs with adaptive sinusoidal activation functions are able to generate frequency-domain wavefield solutions that satisfy wave equations. We also show the flexibility and versatility of the proposed method for various media, including anisotropy, and for models with strong irregular topography.