Kronecker Neural Networks Overcome Spectral Bias for PINN-Based Wavefield Computation

Wavefield computation constitutes the majority of the computational cost for seismic applications, including reverse-time migration and full waveform inversion (FWI). One of the popular approaches is to solve for the wavefields in the frequency domain using the Helmholtz equation. However, the Helmholtz solvers require inversion of a large stiffness matrix that can become computationally intractable for large 3-D models or in the case of modeling high frequencies. Recently, researchers have explored the potential of physics-informed neural networks (PINNs) in solving the Helmholtz equation with limited success. While a number of attractive features have been demonstrated for the PINN-based Helmholtz solvers, their large training cost has been the main impediment in their widespread adoption for wavefield computations. The large training cost is mainly due to the spectral bias of neural networks, which poses difficulty in training the PINN model for high-frequency wavefields. In this work, the author uses Kronecker neural networks (KNNs) that form a general framework for neural networks with adaptive activation functions. He, specifically, implements it using a standard feed-forward neural network using a composite activation function formed using the inverse tangent (atan), exponential linear unit (elu), locally adaptive sine (l-sin), and locally adaptive cosine (l-cos) activation functions. Thanks to the oscillatory noise added by the sine and cosine terms, the network is able to explore more and learn faster. This allows the network to get rid of saturation regions from the output of each layer and overcome slow convergence. Through numerical tests, the efficacy of the proposed approach in fast and accurate wavefield modeling in the frequency domain is demonstrated.