  This work develops neural-network--based preconditioners to accelerate
solution of the Wilson-Dirac normal equation in lattice quantum field theories.
The approach is implemented for the two-flavor lattice Schwinger model near the
critical point. In this system, neural-network preconditioners are found to
accelerate the convergence of the conjugate gradient solver compared with the
solution of unpreconditioned systems or those preconditioned with conventional
approaches based on even-odd or incomplete Cholesky decompositions, as measured
by reductions in the number of iterations and/or complex operations required
for convergence. It is also shown that a preconditioner trained on ensembles
with small lattice volumes can be used to construct preconditioners for
ensembles with many times larger lattice volumes, with minimal degradation of
performance. This volume-transferring technique amortizes the training cost and
presents a pathway towards scaling such preconditioners to lattice field theory
calculations with larger lattice volumes and in four dimensions.
