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

Nathan Moynihan

hep-thFeb 16, 2026

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

Authors: Nathan Moynihan

Organizations: Queen Mary, University of London, Mile End Road, London E1 4NS, UK

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.

Figures & tables

Generate data Evaluate amplitudes

A_{n}(\sigma)

\tilde{A}_{n}(\tilde{\sigma})

, and invariants

\{s_{ij}\}

, on random on-shell phase space points. Split into train/test.

\Downarrow

Build feature sets Either individual amplitudes

A(\sigma_{a})

or bilinears

B_{ab}=A(\sigma_{a})\tilde{A}(\tilde{\sigma}_{b})

\Downarrow

Feature selection Build a feature matrix on the training set and apply CPQR at tolerance

\varepsilon

to expose rank structure and prune redundant features.

\Downarrow

Basis selection Scan left/right amplitude bases and choose a basis that minimises a simple proxy objective (e.g. validation error / model size / spurious-pole indicators).

Figure 1: Schematic of the pipeline used to rediscover KLT-type relations from numerical samples.

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Reconstructing scattering amplitudes from finite, noisy, and mutually inconsistent measurements is an ill-posed inverse problem common to many reactions relevant to particle physics. We introduce S-matrix informed neural networks (SINNs), and demonstrate their ability to learn scattering amplitudes directly from data while respecting first principles. We further develop a novel data selection procedure, which uses the response of constrained neural network ensembles to identify a set of experiments compatible with first principles, and with each other. We apply this framework to

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Abstract

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