Single Reference Frequency Loss for Multi-frequency Wavefield Representation using Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) offer approximate multidimensional functional solutions to the Helmholtz equation that are flexible, require low memory, and have no limitations on the shape of the solution space. However, the neural network (NN) training can be costly and the cost will dramatically increase as we train for multi-frequency wavefields, even if we add frequency to the NN multidimensional function, as the variation of the wavefield with frequency adds complexity to the NN training. Thus, we propose a new loss function for the NN multidimensional input training that will allow us to seamlessly include frequency as a dimension. We specifically utilize the linear relation between frequency and wavenumber (a space representation) to incorporate a reference frequency scaling to the loss function. As a result, the effective wavenumber of the wavefield solution as a function of frequency remains stationary reducing the learning burden on the NN function. We demonstrate the effectiveness of this modified loss function on a layered model.