High-dimensional wavefield solutions based on neural network functions

Wavefield solutions are critical for applications ranging from imaging to full waveform inversion. These wavefields are often large, especially for 3D media, and multiple point sources, like Green's functions. A recently introduced framework based on neural networks admitting functional solutions to partial differential equations (PDEs) offers the opportunity to solve the Helmholtz equation with a neural network (NN) model. The input to such an NN is a location in space and the output are the real and imaginary parts of the scattered wavefield at that location, thus, acting like a function. The network is trained on random input points in space and a variance of the Helmholtz equation for the scattered wavefield is used as the loss function to update the network parameters. In spite of the methods flexibility, like handling irregular surfaces and complex media, and its potential for velocity model building, the cost of training the network far exceeds that of numerical solutions. Relying on the network's ability to learn wavefield features, we extend the dimension of this NN function to learn the wavefield for many sources and frequencies, simultaneously. We show, in this preliminary study, that reasonable wavefield solutions can be predicted using smaller networks. This includes wavefields for frequencies not within the training range. The new NN function has the potential to efficiently represent the wavefield as a function of location in space, as well as source location and frequency.