Neural Spectral Bias and Conformal Correlators I: Introduction and Applications

TL;DR

This paper shows that neural networks can accurately reconstruct conformal field theory correlators using minimal data, leveraging spectral bias and smoothness properties across various theories and dimensions.

Contribution

It introduces a minimal-data neural network approach for CFT correlator reconstruction, highlighting the role of spectral bias and extending to broader kinematic regimes.

Findings

Neural networks accurately reproduce CFT correlators with minimal input data.

The approach is robust across diverse theories and dimensions.

Spectral bias explains the smoothness and reconstructive success of neural networks.

Abstract

We demonstrate that simple feed-forward neural networks (NNs) can accurately compute correlation functions of conformal field theories (CFTs) on a line. Strikingly, by optimising a NN solely on crossing symmetry and providing only the scaling dimension of the leading non-trivial operator and the correlator's value at a single "anchor point", we can reconstruct target physical correlators to within a few percent. We establish the robustness of this minimal-data approach across a broad class of theories and dimensions, including generalised free fields, contact and one-loop Witten diagrams in AdS$_2$, unitary and non-unitary 2d minimal models, the 3d Ising model, and half-BPS correlators in 4d $\mathcal{N}=4$ super-Yang-Mills theory, together with several thermal two-point functions, notably including those of the 3d Ising model. We argue that this remarkable alignment between NNs and CFTs stems from the spectral bias of gradient-based training, which heavily favours smooth functions. To ground this connection, we analyse the smoothness of conformal correlators using fractional Sobolev semi-norms, Chebyshev spectral decompositions, and a measure based on curvature. Finally, we establish the broader reconstructive power of this technique by extending it beyond the diagonal kinematics of the line.