  We describe a procedure for alleviating the fermion sign problem in which
phase fluctuations are explicitly subtracted from the Boltzmann factor. Several
ans\"atze for fluctuations are designed and compared. In the absence of a
sufficiently high-quality ansatz, a neural network can be trained to
parameterize the fluctuations. Demonstrating on the staggered Thirring model in
$1+1$ dimensions, we examine the performance of this method as deeper neural
networks are used, and in conjunction with the well-studied contour deformation
methods.
