  Full Waveform Inversion (FWI) is an inverse problem for estimating the wave
velocity distribution in a given domain, based on observed data on the
boundaries. The inversion is computationally demanding because we are required
to solve multiple forward problems, either in time or frequency domains, to
simulate data that are then iteratively fitted to the observed data. We
consider FWI in the frequency domain, where the Helmholtz equation is used as a
forward model, and its repeated solution is the main computational bottleneck
of the inversion process. To ease this cost, we integrate a learning process of
an encoder-solver preconditioner that is based on convolutional neural networks
(CNNs). The encoder-solver is trained to effectively precondition the
discretized Helmholtz operator given velocity medium parameters. Then, by
re-training the CNN between the iterations of the optimization process, the
encoder-solver is adapted to the iteratively evolving velocity medium as part
of the inversion. Without retraining, the performance of the solver
deteriorates as the medium changes. Using our light retraining procedures, we
obtain the forward simulations effectively throughout the process. We
demonstrate our approach to solving FWI problems using 2D geophysical models
with high-frequency data.
