Multi-variable Integration with a Neural Network

TL;DR

This paper introduces a neural network-based method for automatic integration of parametric integrals over the unit hypercube, achieving high accuracy and significantly faster evaluation than traditional numerical methods.

Contribution

The paper presents a novel neural network approach that fits the primitive of the integrand, enabling fast and accurate integration of complex parametric integrals.

Findings

Achieves per-mil and percent accuracy on example integrals.

Evaluation speed is 40 to 125 times faster than traditional methods.

Method scales favorably with integrand complexity.

Abstract

In this article we present a method for automatic integration of parametric integrals over the unit hypercube using a neural network. The method fits a neural network to the primitive of the integrand using a loss function designed to minimize the difference between multiple derivatives of the network and the function to be integrated. We apply this method to two example integrals resulting from the sector decomposition of a one-loop and two-loop scalar integrals. Our method can achieve per-mil and percent accuracy for these integrals over a range of invariant values. Once the neural network is fitted, the evaluation of the integral is between 40 and 125 times faster than the usual numerical integration method for our examples, and we expect the speed gain to increase with the complexity of the integrand.