Globally Optimal Contour Deformations with Neural Networks

TL;DR

This paper introduces a neural network-based method for selecting optimal contour deformations in Feynman integral evaluations, reducing variance and improving efficiency over heuristic approaches.

Contribution

It presents a novel approach to determine a global contour deformation for entire phase-space regions using minimal initial sampling, enhancing integration accuracy and neural network training.

Findings

Lower variance in integrand with the proposed method

Elimination of retraining neural networks for each phase-space point

Potential for faster and more accurate numerical integration

Abstract

In this article, we explore the use of contour deformation for the numerical evaluation of Feynman integrals after sector decomposition. In existing codes, the contour of integration is determined heuristically for each phase-space point by sampling the integrand. In this work, we introduce a method for choosing the contour deformation for an entire phase-space region using only an initial sampling or training step. We demonstrate that the resulting integrand has a lower variance than that obtained with heuristic methods and show that optimising a contour to reduce the estimated error of a Quasi-Monte Carlo sample is an ill-defined problem. The a priori knowledge of the integration path obtained in this work can be used to improve the speed of conventional integration methods or be leveraged for integration using neural networks, where, crucially, it removes the need to retrain the neural network for each phase-space point. The techniques described in this work can be adapted to other problems where a non-trivial integration path has to be chosen subject to a set of constraints.