Application of a neural network to the sign problem via the path optimization method

Abstract

We introduce the feedforward neural network to attack the sign problem via the path optimization method. The variables of integration is complexified and the integration path is optimized in the complexified space by minimizing the cost function which reflects the seriousness of the sign problem. For the preparation and optimization of the integral path in multi-dimensional systems, we utilize the feedforward neural network. We examine the validity and usefulness of the method in the two-dimensional complex λφ 4 theory at finite chemical potential as an example of the quantum field theory having the sign problem. We show that the average phase factor is significantly enhanced after the optimization and then we can safely perform the hybrid Monte-Carlo method.

Imaginary Chemical Potential, NJL-Type Model and Confinement–Deconfinement Transition

“…Here, we use the analytic continuation from the real to imaginary chemical potentials. The RW periodicity is induced by the factor cos(N c θ) in Equation (8); this factor comes from the baryonic fugacity;…”

“…[1] as an example, but the applicable regions are still limited in µ R /T < 1. It should be noted that there are some attempts to overcome the limitation: famous methods are the complex Langevin method [2,3], the Lefschetz thimble method [4][5][6], the path optimization method [7,8] and so on. However, these methods still have serious problems.…”

A simple approach towards the sign problem using path optimisation

“…Note: While this work was prepared the papers [24][25][26] appeared, in which some similar ideas were presented. Our approach shares with [26] the low computational complexity and the ability to use a relatively small number of parameters, while generalising the ansatz for the contour in a way that enables taking into account the interaction between nearby lattice points.…”

Reweighting Lefschetz Thimbles

“…Recent contributions in the field more generally involve complex manifolds close to Lefschetz thimbles that are optimized such that they ameliorate the sign problem, see e.g. [27][28][29][30].…”