Learning the S-matrix from data: Rediscovering gravity from gauge theory via symbolic regression

TL;DR

This paper shows that symbolic regression can autonomously rediscover fundamental relations in scattering amplitudes, like KLT relations, from numerical data, offering a new data-driven approach to explore the analytic structure of quantum field theories.

Contribution

The paper demonstrates that symbolic regression can recover key amplitude relations directly from data without prior group-theoretic input, advancing data-driven discovery in theoretical physics.

Findings

Successfully rediscovered KLT relations up to five external legs.

Recovered Kleiss--Kuijf and BCJ relations using feature selection.

Established symbolic regression as a practical tool for amplitude analysis.

Abstract

We demonstrate that modern machine-learning methods can autonomously reconstruct several flagship analytic structures in scattering amplitudes directly from numerical on-shell data. In particular, we show that the Kawai--Lewellen--Tye (KLT) relations can be rediscovered using symbolic regression applied to colour-ordered Yang--Mills amplitudes with Mandelstam invariants as input features. Using standard feature-selection techniques, specifically column-pivoted QR factorisation, we simultaneously recover the Kleiss--Kuijf and Bern--Carrasco--Johansson (BCJ) relations, identifying a minimal basis of partial amplitudes without any group-theoretic input. We obtain the tree-level KLT relations with high numerical accuracy up to five external legs, using only minimal theoretical priors, and we comment on the obstacles to generalising the method to higher multiplicity. Our results establish symbolic regression as a practical tool for exploring the analytic structure of the scattering-amplitude landscape, and suggests a general data-driven strategy for uncovering hidden relations in general theories. For comparison, we benchmark this general approach with a recently introduced neural-network based method.