Wavefield reconstruction inversion via machine learned functions

Wavefield reconstruction inversion (WRI) is a PDE constrained optimization problem that aims to mitigate cycle skipping in full-waveform inversion (FWI) among other potential features. WRI is often implemented in the frequency domain, and thus, requires expensive matrix inversions to reconstruct the wavefield. A recently introduced machine learning (ML) framework, called physics-informed neural networks (NNs), is used to predict PDE solutions by setting the physical laws as loss functions. These NNs have shown their effectiveness in solving the Helmholtz equation specifically for the scattered wavefield. By including the recorded data at the sensors’ locations as a data constraint, the NNs can predict the wavefields which simultaneously fit the recorded data and satisfy the Helmholtz equation for a given initial velocity model. Using the predicted wavefields, we build another independent NN to predict the velocity that fits the wavefield. In this new NN, we use spatial coordinates as input to the network, and use the scattered Helmholtz wave to define the loss function. After we train this deep neural network, we are able to predict the velocity in the domain of interest. We demonstrate the potential of the proposed method using a square anomaly model and a simple layered model, and the initial results considering single frequency data show that the ML-based WRI is able to invert for reasonable velocity models.