Hard enforcement of physics-informed neural network solutions of acoustic wave propagation

Simulating the temporal evolution of wavefield solutions through models with heterogeneous material properties is of practical interest for many scientific applications. The acoustic wave equation (AWE) is often used for studying wave propagation in both fluids and solids and is crucial for many applications including seismic imaging and inversion and non-destructive testing. Because analytical AWE solutions rarely exist for complex heterogeneous media, methods for generating numerical AWE solutions are very desirable. Traditional numerical solvers require discrete model representations with many restrictions placed on the shape and spacing of grid elements. This work uses a relatively new class of numerical solvers known as physics-informed neural networks (PINNs) that provide a mesh-free alternative for generating AWE solutions using a deep neural-network framework. We encapsulate a time-domain AWE formulation within a loss function that is used to train network parameters. The initial conditions are implemented by enforcing hard constraints on the neural network instead of including them as separate loss-function terms. We also use a Fourier neural network (FNN) to alleviate the spectral bias commonly observed when using fully connected neural network in the conventional PINN approach. Numerical tests on both 2D homogeneous and heterogeneous velocity models confirm the accuracy of our approach. We observe that using FNNs helps in the convergence of AWE solutions especially for heterogeneous models. We compare PINN-based solutions with those computed by the highly accurate conventional pseudo-spectral method, and observe that the normalized energy differences between the two sets of solutions were less than 4% for all numerical tests.