Single Reference Frequency Loss for Multifrequency Wavefield Representation Using Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) can 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 dramatically increases as we train for multi-frequency wavefields by adding frequency as an additional input to the NN multi-dimensional function. In this case, the often large variation of the wavefield features (specifically wavelength) with frequency adds more complexity to the NN training. Thus, we propose a new loss function for the NN multidimensional input training that allows us to seamlessly include frequency as a dimension. We specifically utilize the linear relation between frequency and wavenumber (the wavefield 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 almost stationary, which reduces the learning burden on the NN function. We demonstrate the effectiveness of this modified loss function on a layered model.