  Wavefield reconstruction inversion (WRI) formulates a PDE-constrained
optimization problem to reduce cycle skipping in full-waveform inversion (FWI).
WRI often requires expensive matrix inversions to reconstruct frequency-domain
wavefields. Physics-informed neural network (PINN) uses the underlying physical
laws as loss functions to train the neural network (NN), and it has shown its
effectiveness in solving the Helmholtz equation and generating Green's
functions, specifically for the scattered wavefield. By including a
data-constrained term in the loss function, the trained NN can reconstruct a
wavefield that simultaneously fits the recorded data and satisfies the
Helmholtz equation for a given initial velocity model. Using the predicted
wavefields, we rely on a small-size NN to predict the velocity using the
reconstructed wavefield. In this velocity prediction NN, spatial coordinates
are used as input data to the network and the scattered Helmholtz equation is
used to define the loss function. After we train this network, we are able to
predict the velocity in the domain of interest. We develop this PINN-based WRI
method and demonstrate its potential using a part of the Sigsbee2A model and a
modified Marmousi model. The results show that the PINN-based WRI is able to
invert for a reasonable velocity with very limited iterations and frequencies,
which can be used in a subsequent FWI application.
