Deep learning of fermion sign fluctuations

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.