Wavefield Solutions from Machine Learned Functions that Approximately Satisfy the Wave Equation

Solving the Helmholtz wave equation provides wavefield solutions that are dimensionally compressed, per frequency, compared to the time domain, which is useful for many applications, like full waveform inversion (FWI). However, the efficiency in attaining such wavefield solutions depends often on the size of the model, which tends to be large at high frequencies and for 3D problems. Thus, we use a recently introduced framework based on predicting such functional solutions through setting the underlying physical equation as a cost function to optimize a neural network for such a task. We specifically seek the solution of the functional scattered wavefield in the frequency domain through a neural network considering a simple homogeneous background model. Feeding the network a reasonable number random points from the model space will ultimately train a fully connected 8-layer deep neural network with each layer having a dimension of 20, to predict the scattered wavefield function. Initial tests on a two-box-shaped scatterer model with a source in the middle, as well as, a layered model with a source on the surface demonstrate the successful training of the NN for this application and provide us with a peek into the potential of such an approach.