Helmholtz-equation solution in nonsmooth media by a physics-informed neural network incorporating quadratic terms and a perfectly matching layer condition

Abstract

Frequency-domain simulation of seismic waves plays an important role in seismic inversion, but it remains challenging in large models. The recently proposed physics-informed neural network (PINN), as an effective deep-learning method, has achieved successful applications in solving a wide range of partial differential equations (PDEs), and there is still room for improvement on this front. For example, PINN can lead to inaccurate solutions when PDE coefficients are nonsmooth and describe structurally complex media. Thus, we solve the acoustic and visco-acoustic scattered-field (Lippmann-Schwinger) wave equation in the frequency domain with PINN instead of the wave equation to remove the source singularity. We first illustrate that nonsmooth velocity models lead to inaccurate wavefields when no boundary conditions are implemented in the loss function. Then, we add the perfectly matched layer (PML) conditions in the loss function to better couple the real and imaginary parts of the wavefield. Moreover, we design new neurons by replacing the classical affine function with a quadratic function in the argument of the activation function to better capture the nonsmooth features of the wavefields. We find that the PML condition and the quadratic functions improve the results including handling attenuation and discuss the reason for this improvement. We also illustrate that a network trained to predict a wavefield for a specific medium can be used as an initial model of the neural network for predicting other wavefields corresponding to PDE-coefficient alterations and improving the convergence speed accordingly. This pretraining strategy should find applications in iterative full-waveform inversion and time-lag target-oriented imaging when the model perturbation between two consecutive iterations or two consecutive experiments is small.