A machine learning pipeline for autonomous numerical analytic continuation of Dyson-Schwinger equations

TL;DR

This paper proposes a machine learning pipeline utilizing deep learning and reinforcement learning to autonomously detect singularities and optimize contour deformations in solving Dyson-Schwinger equations in the complex domain.

Contribution

It introduces a novel ML-based method to automatically identify poles and branch cuts and suggest contour deformations, improving the efficiency of solving Dyson-Schwinger equations.

Findings

Proof of concept for pole detection

Proof of concept for branch cut detection

Initial results on contour deformation suggestions

Abstract

Dyson-Schwinger equations (DSEs) are a non-perturbative way to express n-point functions in quantum field theory. Working in Euclidean space and in Landau gauge, for example, one can study the quark propagator Dyson-Schwinger equation in the real and complex domain, given that a suitable and tractable truncation has been found. When aiming for solving these equations in the complex domain, that is, for complex external momenta, one has to deform the integration contour of the radial component in the complex plane of the loop momentum expressed in hyper-spherical coordinates. This has to be done in order to avoid poles and branch cuts in the integrand of the self-energy loop. Since the nature of Dyson-Schwinger equations is such, that they have to be solved in a self-consistent way, one cannot analyze the analytic properties of the integrand after every iteration step, as this would not be feasible. In these proceedings, we suggest a machine learning pipeline based on deep learning (DL) approaches to computer vision (CV), as well as deep reinforcement learning (DRL), that could solve this problem autonomously by detecting poles and branch cuts in the numerical integrand after every iteration step and by suggesting suitable integration contour deformations that avoid these obstructions. We sketch out a proof of principle for both of these tasks, that is, the pole and branch cut detection, as well as the contour deformation.