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arxiv: 2604.18673 · v1 · submitted 2026-04-20 · ✦ hep-th

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Neural Networks Reveal a Universal Bias in Conformal Correlators

Kausik Ghosh , Sidhaarth Kumar , Vasilis Niarchos , Andreas Stergiou
Authors on Pith no claims yet

Pith reviewed 2026-05-10 04:27 UTC · model grok-4.3

classification ✦ hep-th
keywords neural networksconformal field theorycrossing symmetrycorrelatorsspectral biasnon-perturbative QFTvariational methods
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The pith

Simple neural networks trained on crossing symmetry reconstruct conformal correlators from minimal input data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that basic neural networks, when trained to enforce crossing symmetry, can recover conformal correlators restricted to a line with high accuracy using only an external scaling dimension, a spectral gap, and the correlator value at one point. This holds across many conformal theories and spacetime dimensions for both four-point functions and thermal two-point functions. The authors link the result to a match between the spectral bias of gradient descent training and an intrinsic smoothness property of conformal field theory, which in turn points toward a variational principle that could serve as a new computational tool for non-perturbative quantum field theory.

Core claim

We propose that simple neural networks (NNs) trained on crossing symmetry can reconstruct conformal correlators restricted to a line to remarkable accuracy. The input is minimal: an external scaling dimension, a spectral gap, and the value of the correlator at a single point. We present evidence across a wide range of conformal theories and dimensions, for both four-point and thermal two-point functions. We attribute these observations to the spectral bias of gradient-based NN training, which appears to align with an intrinsic smoothness property of conformal field theory. This suggests a novel variational principle for conformal correlators and opens a path towards a powerful new framework.

What carries the argument

Gradient-based neural network training whose spectral bias aligns with the smoothness property of conformal correlators, allowing reconstruction from crossing symmetry plus minimal data.

If this is right

  • The method provides a variational principle that computes conformal correlators without solving the full crossing equations directly.
  • Reconstruction accuracy persists across different spacetime dimensions and for both four-point and thermal two-point functions.
  • Only an external scaling dimension, spectral gap, and one correlator value are required as input.
  • The approach opens a computational route to non-perturbative quantities in quantum field theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The observed alignment may indicate that conformal data possess a hidden low-frequency structure that gradient descent naturally discovers.
  • Similar networks could be tested on other constraints such as unitarity bounds or modular invariance to see whether the same bias operates.
  • If the smoothness match is universal, the technique might extend to correlators in non-conformal theories with analogous analytic properties.

Load-bearing premise

The spectral bias that arises in gradient-based neural network training aligns with an intrinsic smoothness property of conformal field theory.

What would settle it

Train the network on crossing symmetry with the stated minimal inputs and check whether reconstruction error remains low for a known exact correlator such as the Ising four-point function; a sharp rise in error would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.18673 by Andreas Stergiou, Kausik Ghosh, Sidhaarth Kumar, Vasilis Niarchos.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Thermal two-point function [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Collected plots of anchored neural network predictions against analytic results. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

We propose that simple neural networks (NNs) trained on crossing symmetry can reconstruct conformal correlators restricted to a line to remarkable accuracy. The input is minimal: an external scaling dimension, a spectral gap, and the value of the correlator at a single point. We present evidence across a wide range of conformal theories and dimensions, for both four-point and thermal two-point functions. We attribute these observations to the spectral bias of gradient-based NN training, which appears to align with an intrinsic smoothness property of conformal field theory. This suggests a novel variational principle for conformal correlators and opens a path towards a powerful new computational framework for non-perturbative quantum field theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes that simple neural networks trained to enforce crossing symmetry can reconstruct conformal correlators restricted to a line to high accuracy. The inputs consist of an external scaling dimension, a spectral gap, and the correlator value at a single point. Evidence is claimed across a range of CFTs and dimensions for both four-point functions and thermal two-point functions. The success is attributed to the spectral bias of gradient-based NN training aligning with an intrinsic smoothness property of CFTs, suggesting a novel variational principle for conformal correlators.

Significance. If the central claims are substantiated, the work could introduce a new variational framework for computing non-perturbative CFT correlators with minimal input data, potentially offering computational advantages in regimes where bootstrap or perturbative methods are limited. The interdisciplinary link between NN training dynamics and CFT structure would be noteworthy if isolated from generic approximation effects.

major comments (3)
  1. [Abstract] Abstract: The claims of 'remarkable accuracy' and 'evidence across a wide range' are not accompanied by quantitative metrics (e.g., mean squared error, maximum deviation, or error bars), training protocols, data selection criteria, or network hyperparameters, preventing verification of the central reconstruction results.
  2. [Results] Results section: The attribution of accuracy to spectral bias in gradient-based training lacks isolating controls; no comparisons are presented to other approximators (such as Chebyshev polynomials or Fourier bases) that minimize the identical crossing-violation loss on the same low-dimensional inputs, leaving open whether the outcome stems from NN-specific bias or from the smoothness of crossing-constrained 1D functions.
  3. [Methods] Methods: The formulation of the loss function used to enforce crossing symmetry during training is not specified in sufficient detail (e.g., how the crossing equation is discretized or weighted), which is load-bearing for reproducing the reported reconstructions and for assessing whether the procedure is parameter-free as implied.
minor comments (2)
  1. Clarify the precise neural network architecture (number of layers, neurons per layer, activation functions) and the range of external dimensions and gaps explored in the numerical experiments.
  2. [Discussion] The discussion of the 'universal bias' would benefit from explicit comparison to the established spectral bias literature cited, to distinguish the CFT-specific alignment from general ML observations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address each major comment below and will revise the manuscript to improve reproducibility, add controls, and clarify methodological details.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claims of 'remarkable accuracy' and 'evidence across a wide range' are not accompanied by quantitative metrics (e.g., mean squared error, maximum deviation, or error bars), training protocols, data selection criteria, or network hyperparameters, preventing verification of the central reconstruction results.

    Authors: We agree that quantitative metrics and experimental details are necessary for verification. In the revised manuscript we will add a new table in the Results section reporting mean squared errors, maximum deviations, and error bars (from multiple random seeds) for each CFT and dimension considered. We will also include a supplementary section listing the exact network architectures, learning rates, epoch counts, data selection criteria, and training protocols used throughout the study. revision: yes

  2. Referee: [Results] Results section: The attribution of accuracy to spectral bias in gradient-based training lacks isolating controls; no comparisons are presented to other approximators (such as Chebyshev polynomials or Fourier bases) that minimize the identical crossing-violation loss on the same low-dimensional inputs, leaving open whether the outcome stems from NN-specific bias or from the smoothness of crossing-constrained 1D functions.

    Authors: This criticism is well-taken; the present manuscript does not contain such isolating controls. In the revision we will add direct comparisons in which Chebyshev polynomials and Fourier bases are optimized to minimize the identical crossing-violation loss on the same minimal inputs. These controls will quantify whether the reported accuracy is attributable to the spectral bias of gradient-based NN training or is a generic consequence of smoothness under crossing constraints. revision: yes

  3. Referee: [Methods] Methods: The formulation of the loss function used to enforce crossing symmetry during training is not specified in sufficient detail (e.g., how the crossing equation is discretized or weighted), which is load-bearing for reproducing the reported reconstructions and for assessing whether the procedure is parameter-free as implied.

    Authors: We acknowledge the omission. The loss is the mean-squared violation of the crossing equation, evaluated on a uniform grid of 200 points in z ∈ [0,1] with equal weighting. The procedure depends on this discretization choice and is therefore not strictly parameter-free. In the revised Methods section we will state the precise loss functional, the grid size, the weighting scheme, and a brief sensitivity analysis with respect to grid resolution. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical NN reconstruction uses external crossing constraint as independent input.

full rationale

The paper's derivation chain consists of training NNs to minimize a crossing-violation loss on functions parameterized by external scaling dimension, spectral gap, and one-point value, then observing accurate reconstruction of known CFT correlators. This is not self-definitional or a fitted input renamed as prediction, because crossing symmetry is an external CFT axiom enforced during training rather than derived from the NN outputs. Attribution to spectral bias references independent machine-learning literature on gradient descent preferences, without reducing to a self-citation chain or ansatz smuggled from prior author work. No uniqueness theorem or renaming of known results is invoked as load-bearing. The central claim remains an empirical observation verifiable against external CFT data and is self-contained against benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two standard domain assumptions from CFT and machine learning; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Crossing symmetry is a fundamental constraint on conformal correlators
    Standard axiom of conformal field theory invoked to train the networks.
  • domain assumption Gradient-based neural network training exhibits spectral bias toward smooth functions
    Established property of neural networks used to explain the observed accuracy.

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Forward citations

Cited by 2 Pith papers

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  2. Neural Networks, Dispersion Relations and the Thermal Bootstrap

    hep-th 2026-05 unverdicted novelty 4.0

    A neural-network approach with dispersion relations handles infinite OPE towers in thermal conformal correlators without positivity.

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