Learning Feynman integrals from differential equations with neural networks

TL;DR

This paper explores a novel physics-informed deep learning method to numerically evaluate Feynman integrals by training neural networks to solve their differential equations, offering rapid evaluation without relying on canonical forms or large datasets.

Contribution

It introduces a new neural network-based approach for solving Feynman integrals via differential equations, bypassing traditional analytical bottlenecks and dataset requirements.

Findings

Achieved around 1% relative difference in two-loop Feynman integrals.

Training times are approximately one hour on a laptop GPU.

Method provides rapid, approximate evaluations of Feynman integrals.

Abstract

We perform an exploratory study of a new approach for evaluating Feynman integrals numerically. We apply the recently-proposed framework of physics-informed deep learning to train neural networks to approximate the solution to the differential equations satisfied by the Feynman integrals. This approach relies neither on a canonical form of the differential equations, which is often a bottleneck for the analytical techniques, nor on the availability of a large dataset, and after training yields essentially instantaneous evaluation times. We provide a proof-of-concept implementation within the PyTorch framework, and apply it to a number of one- and two-loop examples, achieving a mean magnitude of relative difference of around 1% at two loops in the physical phase space with network training times on the order of an hour on a laptop GPU.