# Gauge-Equivariant Graph Neural Networks for Lattice Gauge Theories

arXiv: https://arxiv.org/abs/2604.20797 · HTML: https://mlsanity.com/paper/2604.20797/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 67 references that are in the map.
- First posted on arXiv 22 Apr 2026.

## Builds on (most cited first, 25 of 25)

- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentials](https://mlsanity.com/paper/2205.06643/) (2022)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [Machine Learning in the Search for New Fundamental Physics](https://mlsanity.com/paper/2112.03769/) (2021)
- [Lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2012.12901/) (2020)
- [Gauge-equivariant flow models for sampling in lattice field theories with pseudofermions](https://mlsanity.com/paper/2207.08945/) (2022)
- [Gauge equivariant neural networks for quantum lattice gauge theories](https://mlsanity.com/paper/2012.05232/) (2020)
- [Equivariant Networks for Crystal Structures](https://mlsanity.com/paper/2211.15420/) (2022)
- [Machine Learning for Condensed Matter Physics](https://mlsanity.com/paper/2005.14228/) (2020)
- [Gauge Invariant and Anyonic Symmetric Transformer and RNN Quantum States for Quantum Lattice Models](https://mlsanity.com/paper/2101.07243/) (2021)
- [Generalization capabilities of translationally equivariant neural networks](https://mlsanity.com/paper/2103.14686/) (2021)
- [Gauge-equivariant neural networks as preconditioners in lattice QCD](https://mlsanity.com/paper/2302.05419/) (2023)
- [Gauge-equivariant pooling layers for preconditioners in lattice QCD](https://mlsanity.com/paper/2304.10438/) (2023)
- [Deep learning lattice gauge theories](https://mlsanity.com/paper/2405.14830/) (2024)

## Built on by (most cited first, 3 of 3)

- [Theoretical Aspects of Lie Groupoid and Lie Algebroid Equivariant Convolutional Neural Networks](https://mlsanity.com/paper/2606.02758/) (2026)
- [Adjusted Cup-Product Neural Layer](https://mlsanity.com/paper/2606.13568/) (2026)
- [Emergent Loop Geometry in Gauge-Equivariant Graph Neural Networks](https://mlsanity.com/paper/2610.18742/) (2026)

## Abstract (arXiv)

Local gauge symmetry underlies fundamental interactions and strongly correlated quantum matter, yet existing machine-learning approaches lack a general, principled framework for learning under site-dependent symmetries, particularly for intrinsically nonlocal observables. Here we introduce a gauge-equivariant graph neural network that embeds non-Abelian symmetry directly into message passing via matrix-valued, gauge-covariant features and symmetry-compatible updates, extending equivariant learning from global to fully local symmetries. In this formulation, message passing implements gauge-covariant transport across the lattice, allowing nonlocal correlations and loop-like structures to emerge naturally from local operations. We validate the approach across pure gauge, gauge-matter, and dynamical regimes, establishing gauge-equivariant message passing as a general paradigm for learning in systems governed by local symmetry.