# Gauge-equivariant neural networks as preconditioners in lattice QCD

arXiv: https://arxiv.org/abs/2302.05419 · HTML: https://mlsanity.com/paper/2302.05419/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 39 references that are in the map.
- First posted on arXiv 10 Feb 2023.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [Learning to Optimize Multigrid PDE Solvers](https://mlsanity.com/paper/1902.10248/) (2019)
- [Learning Algebraic Multigrid Using Graph Neural Networks](https://mlsanity.com/paper/2003.05744/) (2020)
- [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)
- [Learning optimal multigrid smoothers via neural networks](https://mlsanity.com/paper/2102.12071/) (2021)
- [MGIC: Multigrid-in-Channels Neural Network Architectures](https://mlsanity.com/paper/2011.09128/) (2020)
- [MGiaD: Multigrid in all dimensions. Efficiency and robustness by coarsening in resolution and channel dimensions](https://mlsanity.com/paper/2211.05525/) (2022)

## Built on by (most cited first, 9 of 9)

- [Advances in machine-learning-based sampling motivated by lattice quantum chromodynamics](https://mlsanity.com/paper/2309.01156/) (2023)
- [Geometrical aspects of lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2303.11448/) (2023)
- [Gauge-equivariant pooling layers for preconditioners in lattice QCD](https://mlsanity.com/paper/2304.10438/) (2023)
- [Gauge-Equivariant Graph Neural Networks for Lattice Gauge Theories](https://mlsanity.com/paper/2604.20797/) (2026)
- [Machine learning a fixed point action for SU(3) gauge theory with a gauge equivariant convolutional neural network](https://mlsanity.com/paper/2401.06481/) (2024)
- [Graph Neural Networks in the Wilson Loop Representation of Abelian Lattice Gauge Theories](https://mlsanity.com/paper/2605.03901/) (2026)
- [Graph Lineages and Skeletal Graph Products](https://mlsanity.com/paper/2508.00197/) (2025)
- [Matrix-free Neural Preconditioner for the Dirac Operator in Lattice Gauge Theory](https://mlsanity.com/paper/2509.10378/) (2025)
- [Emergent Loop Geometry in Gauge-Equivariant Graph Neural Networks](https://mlsanity.com/paper/2610.18742/) (2026)

## Abstract (arXiv)

We demonstrate that a state-of-the art multi-grid preconditioner can be learned efficiently by gauge-equivariant neural networks. We show that the models require minimal re-training on different gauge configurations of the same gauge ensemble and to a large extent remain efficient under modest modifications of ensemble parameters. We also demonstrate that important paradigms such as communication avoidance are straightforward to implement in this framework.