Physics-informed neural wavefields with Gabor basis functions

Abstract: Recently, Physics-Informed Neural Networks (PINNs) have gained significant attention for their versatile interpolation capabilities in solving partial differential equations (PDEs). Despite their potential, the training can be computationally demanding, especially for intricate functions like wavefields. This is primarily due to the neural-based (learned) basis functions, biased toward low frequencies, as they are dominated by polynomial calculations, which are not inherently wavefield-friendly. In response, we propose an approach to enhance the efficiency and accuracy of neural network wavefield solutions by modeling them as linear combinations of Gabor basis functions that satisfy the wave equation. Specifically, for the Helmholtz equation, we augment the fully connected neural network model with an adaptable Gabor layer constituting the final hidden layer, employing a weighted summation of these Gabor neurons to compute the predictions (output). These weights/coefficients of the Gabor functions are learned from the previous hidden layers that include nonlinear activation functions. To ensure the Gabor layer's utilization across the model space, we incorporate a smaller auxiliary network to forecast the center of each Gabor function based on input coordinates. Realistic assessments showcase the efficacy of this novel implementation compared to the vanilla PINN, particularly in scenarios involving high-frequencies and realistic models that are often challenging for PINNs.