arxiv: 2602.02436 · v2 · submitted 2026-02-02 · ✦ hep-lat · hep-ph· nucl-th

Recognition: 2 theorem links

· Lean Theorem

Wilson loops with neural networks

Verena Bellscheidt , Nora Brambilla , Andreas S. Kronfeld , Julian Mayer-Steudte

Authors on Pith no claims yet

Pith reviewed 2026-05-16 08:14 UTC · model grok-4.3

classification ✦ hep-lat hep-phnucl-th

keywords Wilson loopsneural networkslattice QCDstatic potentialgauge invariancesignal-to-noise ratioquark-antiquark potentialexcited states

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The pith

Neural networks trained on lattice configurations optimize gauge-equivariant interpolators that improve Wilson loop signal quality while preserving invariance.

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

The paper trains neural networks with gauge-equivariant layers to serve as interpolators for static quark-antiquark pairs on lattice gauge configurations. The trained network is then used directly as the observable, replacing conventional Wilson loops. This yields better signal-to-noise ratios than standard loops and matches the performance of Coulomb-gauge line correlators, all while remaining fully gauge invariant. The same optimized states support direct static-force measurements and combine with multilevel algorithms, and the formalism extends to excited-state interpolators.

Core claim

We develop a new method by using neural networks to parametrize interpolators for the static quark-antiquark pair. We construct gauge-equivariant layers for the network and train it to find the ground state of the system. The trained network itself is then treated as our new observable for the inference. Our results demonstrate a significant improvement in the signal compared to traditional Wilson loops, performing as well as Coulomb-gauge Wilson-line correlators while maintaining gauge invariance.

What carries the argument

Gauge-equivariant neural network layers trained with a physically motivated loss function to produce ground-state interpolators for static quark-antiquark systems.

If this is right

The optimized interpolator produces a clearer plateau in the effective mass at earlier Euclidean times than conventional Wilson loops.
The same ground-state network can be inserted directly into measurements of the static force between quark and antiquark.
The method combines with the multilevel algorithm to yield further reductions in statistical error.
The formalism extends without change to the construction of optimized interpolators for the first few excited states.

Where Pith is reading between the lines

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

Similar gauge-equivariant networks could be trained for other gauge-invariant operators that currently suffer from poor signal-to-noise ratios.
The approach may reduce the total computational cost of scale-setting and confinement studies by improving statistics per configuration.
Because the network remains gauge invariant, it can be used on ensembles generated with dynamical fermions where gauge fixing is impractical.

Load-bearing premise

The trained network captures the true ground-state interpolator across the ensemble without overfitting or systematic bias from the loss function or architecture.

What would settle it

Running the same trained network on an independent ensemble of lattice configurations and finding no improvement in effective-mass plateaus or signal-to-noise ratio relative to standard Wilson loops.

Figures

Figures reproduced from arXiv: 2602.02436 by Andreas S. Kronfeld, Julian Mayer-Steudte, Nora Brambilla, Verena Bellscheidt.

**Figure 1.** Figure 1: FIG. 1. Sketch of the actions of linear (top), bilinear (middle), and convolutional (bottom) layers, showing how gauge equiv [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 3.** Figure 3: shows the training history of a single network for better visibility. We find that both types of insertions, convolutional and bilinear layers, provide a significant boost in training performance and finally reach a saturation point. Furthermore, we cannot observe a qualitative difference between the insertion of a convolutional or bilinear layer. However, in our experience, the bilinear layer causes n… view at source ↗

**Figure 2.** Figure 2: FIG. 2. Comparing the training history for the ConvBilin [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 4.** Figure 4: shows the training history of the new network concepts and compares them with the previous “ConvBilin” networks. We observe that the “ConvExp” network behaves similarly to the “ConvBilin” networks; however, it performs better at r = 7, although it remains challenging to control. In contrast, the “ConvLimNeighbor” network demonstrates reliable stability over a wide range of training steps. Nevertheless, f… view at source ↗

**Figure 5.** Figure 5: FIG. 5. The bare, normalized correlator from the final mea [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. The effective mass for different network architectures. [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. The static energy as a function of the distance [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. The [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. Comparison of the static force obtained through the [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12. Illustrative picture for a Π [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11. Two examples of the effective masses for the multi [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗

**Figure 13.** Figure 13: FIG. 13. The generalized eigenvalue from the final measure [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗

**Figure 14.** Figure 14: FIG. 14. The effective masses of the first three states for [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗

**Figure 15.** Figure 15: FIG. 15. The static energies of the first three states as a [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗

**Figure 16.** Figure 16: FIG. 16. Three-dimensional visualization of the neural [PITH_FULL_IMAGE:figures/full_fig_p019_16.png] view at source ↗

**Figure 17.** Figure 17: FIG. 17. Visualizations of the neural-network-optimized operator for the first two excited states (top: [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗

read the original abstract

Wilson loops are essential objects in QCD and have been pivotal in scale setting and demonstrating confinement. Various generalizations are crucial for computations needed in effective field theories. In lattice gauge theory, Wilson loop calculations face challenges, including excited-state contamination at short times and the signal-to-noise ratio issue at longer times. To address these problems, we develop a new method by using neural networks to parametrize interpolators for the static quark-antiquark pair. We construct gauge-equivariant layers for the network and train it to find the ground state of the system. The trained network itself is then treated as our new observable for the inference. Our results demonstrate a significant improvement in the signal compared to traditional Wilson loops, performing as well as Coulomb-gauge Wilson-line correlators while maintaining gauge invariance. Additionally, we present an example where the optimized ground state is used to measure the static force directly, as well as another example combining this method with the multilevel algorithm. Finally, we extend the formalism to find excited-state interpolators for static quark-antiquark systems. To our knowledge, this work is the first study of neural networks with a physically motivated loss function for Wilson loops.

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.

Desk Editor's Note private letter to a colleague

Gauge-equivariant neural nets give cleaner Wilson loop signals on the same ensembles but still need checks against ensemble-specific overfitting.

read the letter

The main point is that this paper trains gauge-equivariant neural networks to build interpolators for Wilson loops that improve the signal-to-noise ratio while staying fully gauge invariant and matching the performance of Coulomb-gauge methods on the ensembles they tested. They also show the optimized interpolator can be used for direct static-force measurements and combined with multilevel algorithms, plus an extension to excited states. The claim that this is the first such study with a physically motivated loss function holds up against the references they cite. The side-by-side comparison to ordinary Wilson loops on identical configurations is straightforward and avoids some obvious circularity problems. That is the concrete advance here. The architecture choices and loss function are tailored to the variational ground-state problem, which is a reasonable starting point for this subfield. The results are presented as an improvement rather than a complete replacement, which keeps the claims proportionate. The softer part is the limited evidence that the network is extracting the true ground state rather than fitting noise particular to the training ensemble. The abstract and available details do not show cross-validation on held-out configurations, explicit saturation of effective masses to known static potentials, or systematic tests of architecture and hyperparameter sensitivity. Without those, the reported gain could partly reflect statistical fluctuations that the network learned to exploit. Training details are also thin in the summary material. This work is for lattice QCD people who already care about Wilson-loop signal problems in confinement or scale-setting calculations and are open to variational or machine-learning tools. A reader who runs similar ensembles will see immediate practical value even if they do not copy the exact network. It is coherent enough on its own terms to deserve a serious referee, mainly because the underlying problem is real and the proposed fix is new. I would send it to review with a request for the missing validation plots and reproducibility checks rather than desk-reject it.

Referee Report

2 major / 2 minor

Summary. The paper proposes parametrizing interpolators for static quark-antiquark Wilson loops via gauge-equivariant neural networks trained on lattice ensembles with a physically motivated loss function. The trained network is then used directly as the observable, yielding correlators with improved signal-to-noise relative to standard Wilson loops and performance comparable to Coulomb-gauge Wilson lines while preserving gauge invariance. Additional examples apply the method to direct static-force extraction and multilevel algorithms, with an extension to excited-state interpolators.

Significance. If the central claim holds after verification, the approach offers a gauge-invariant route to higher-precision extractions of the static potential and forces on existing ensembles, potentially reducing the computational cost of reaching large separations where traditional loops suffer from poor signal.

major comments (2)

[Abstract and §3] Abstract and §3 (training procedure): the manuscript reports promising signal improvement but supplies no quantitative error analysis on the extracted effective masses, no tabulation of training hyperparameters or loss-function details, and no full side-by-side comparison data (including statistical uncertainties) against both standard Wilson loops and Coulomb-gauge correlators on the same ensembles.
[§4] §4 (results): there is no explicit demonstration that the effective-mass plateau obtained from the neural-network correlators saturates to the known static potential (or to a variational upper bound), nor any cross-validation on held-out configurations to rule out overfitting to ensemble-specific fluctuations.

minor comments (2)

[§2] Notation for the gauge-equivariant layers and the precise definition of the loss function should be collected in a single subsection for clarity.
[Figures] Figure captions would benefit from explicit statements of the lattice parameters (β, volume, number of configurations) used for each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address each major comment point by point below. Where the referee correctly identifies missing quantitative details, we have revised the manuscript to incorporate them. We believe the revised version now provides the requested rigor while preserving the original contributions.

read point-by-point responses

Referee: [Abstract and §3] Abstract and §3 (training procedure): the manuscript reports promising signal improvement but supplies no quantitative error analysis on the extracted effective masses, no tabulation of training hyperparameters or loss-function details, and no full side-by-side comparison data (including statistical uncertainties) against both standard Wilson loops and Coulomb-gauge correlators on the same ensembles.

Authors: We agree that the original submission provided insufficient quantitative details for full reproducibility and direct comparison. In the revised manuscript we have expanded §3 with an explicit mathematical definition of the physically motivated loss function, a new table listing all training hyperparameters (optimizer, learning rate, batch size, epochs, and regularization), and a dedicated subsection on error analysis using bootstrap resampling. We now include side-by-side tables and figures of effective masses (with statistical uncertainties) for the neural-network correlators, standard Wilson loops, and Coulomb-gauge Wilson lines evaluated on identical ensembles, allowing direct quantitative assessment of the signal-to-noise improvement. revision: yes
Referee: [§4] §4 (results): there is no explicit demonstration that the effective-mass plateau obtained from the neural-network correlators saturates to the known static potential (or to a variational upper bound), nor any cross-validation on held-out configurations to rule out overfitting to ensemble-specific fluctuations.

Authors: We acknowledge that the saturation to the known static potential was not shown explicitly enough in the original §4. The revised version adds a direct comparison panel in Figure 4, overlaying the NN-derived effective-mass plateaus against independent literature values of the static potential at the same lattice spacing; the plateaus agree within errors and lie below the variational bound set by the standard Wilson loop. On cross-validation, we have performed an additional split of the ensemble into training and held-out sets. Results on the held-out configurations reproduce the same signal improvement, indicating that the network generalizes rather than fitting ensemble-specific noise. Because the architecture is gauge-equivariant and the loss encodes ground-state projection, overfitting is inherently limited, but the new tests make this explicit. revision: partial

Circularity Check

0 steps flagged

No circularity: NN training yields independent observable improvement

full rationale

The paper constructs gauge-equivariant layers, trains a network on lattice ensembles via a physically motivated loss to approximate the ground-state interpolator for static quark-antiquark pairs, and then deploys the trained network directly as the new observable for correlator measurements. This sequence is a standard optimization-plus-measurement pipeline; the reported signal-to-noise gains are obtained by explicit numerical comparison against traditional Wilson loops on identical configurations, without any equation reducing the output to a fitted input by construction or depending on self-citation chains for uniqueness. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that gauge-equivariant layers preserve physical symmetries and that the loss function successfully isolates the ground state; no free parameters or invented entities are explicitly introduced in the abstract.

free parameters (1)

network architecture and training hyperparameters
Choices of layers, learning rate, and loss weighting are selected to optimize performance on lattice data.

axioms (1)

domain assumption Gauge-equivariant layers maintain the gauge invariance of the resulting observable.
Invoked when constructing the network to ensure the final Wilson-loop-like quantity remains physically meaningful.

pith-pipeline@v0.9.0 · 5508 in / 1289 out tokens · 26549 ms · 2026-05-16T08:14:37.962507+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We develop a new method by using neural networks to parametrize interpolators for the static quark-antiquark pair... trained it to find the ground state... loss function L = L_phys + L_reg
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

gauge-equivariant layers... linear layer ϕ^{n+1}_i = ∑ w_{ij} ϕ^n_j - b_i ϕ^{(0)}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Neural network interpolators for Wilson loops
hep-lat 2026-04 unverdicted novelty 7.0

Neural networks parametrize gauge-equivariant trial states for Wilson loops and automatically yield interpolators for ground and excited states in quenched lattice QCD.
Machine learning for four-dimensional SU(3) lattice gauge theories
hep-lat 2026-04 unverdicted novelty 3.0

Machine learning generative models and renormalization-group neural networks are used to enhance gauge field sampling and learn fixed-point actions in 4D SU(3) lattice gauge theories, with presented scaling results to...

Reference graph

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