Machine learned Green's functions that approximately satisfy the wave equation

Green’s functions are wavefield solutions for a particular point source. They form basis functions to build wavefields for modeling and inversion. However, calculating Green’s functions are both costly and memory intensive. We formulate frequency-domain machine-learned Green’s functions that are represented by neural networks (NN). This NN outputs a complex number (two values representing the real and imaginary part) for the scattered Green’s function at a location in space for a specific source location (both locations are input to the network). Considering a background homogeneous medium admitting an analytical Green’s function solution, the network is trained by fitting the output perturbed Green’s function and its derivatives to the wave equation expressed in terms of the perturbed Green’s function. The derivatives are calculated through the concept of automatic differentiation. In this case, the background Green’s function absorbs the point source singularity, which will allow us to train the network using random points over space and source location using a uniform distribution. Thus, feeding a reasonable number of random points from the model space will ultimately train a fully connected 8-layer deep neural network, to predict the scattered Green’s function. Initial tests on part of the simple layered model (extracted from the left side of the Marmousi model) with sources on the surface demonstrate the successful training of the NN for this application. Using the trained NN model for the Marmousi as an initial NN model for solving for the scattered Green’s function for a 2D slice from the Sigsbee model helped the NN converge faster to a reasonable solution