A Machine Learning Approach for Lattice Gauge Fixing

TL;DR

A novel machine learning framework for lattice gauge fixing is presented, where Wilson lines are utilized to construct gauge transformation matrices within a convolutional neural network, and the model parameters are optimized via backpropagation.

Abstract

Gauge fixing is an essential step in lattice QCD calculations, particularly for studying gauge-dependent observables. Traditional iterative algorithms are computationally expensive and often suffer from critical slowing down and scaling bottlenecks on large lattices. We present a novel machine learning framework for lattice gauge fixing, where Wilson lines are utilized to construct gauge transformation matrices within a convolutional neural network. The model parameters are optimized via backpropagation, and we introduce a hybrid strategy that combines a neural-network-based transformation with subsequent iterative methods. Preliminary tests on SU(3) gauge theory ensembles for Coulomb gauge demonstrate the potential of this approach to improve the efficiency of lattice gauge fixing. Furthermore, we show that the model exhibits lattice size transferability, where parameters optimized on smaller lattices remain effective for larger volumes without additional training. This framework provides a scalable path toward mitigating critical slowing down in high-precision gauge fixing.

Figures (4)

• Figure 1: Training history of the L21S2 CNN, consisting of 21 layers with Wilson lines of lengths one and two. The blue curves represent the L21S2-1N-Z scheme (small dataset), and the orange curves denote the L21S2-4N-Z scheme (large dataset).
• Figure 2: Training history of the L12S3 CNN, featuring 12 layers and Wilson lines up to length three. The blue curves denote the L12S3-1N-Z scheme (small dataset), while the orange curves represent L12S3-4N-Z (large dataset). The green curves illustrate the incremental training scheme (L12S3-4N-W), where training resumed from the 14th epoch of the small-set run using the larger dataset. Its large value of $|\Delta \mathscr{F}|$ at the beginning reflects the reconfiguration of the model as it adapts to the richer dataset, eventually aligning with L12S3-4N-Z.
• Figure 3: Gauge-fixing performance for the RC32x48 ensemble across 100 test configurations. Comparisons are shown between the pure iterative baseline (blue, beginning with 250 LA steps) and hybrid schemes using L21S2-1N-Z (orange) and L12S4-4N-W (green) trained parameter. The panels display: (Top) Evolution of the gauge-fixing functional $F[g]$, with solid lines indicating mean values and bands representing standard deviations. (Middle) Relative difference $\Delta F[g]$ between successive iterations; bands denote the range between maximum and minimum values across incomplete configurations. (Bottom) The number of incomplete configurations remaining throughout the SD iterations. Normalized total computational costs are provided in the legend, representing the area under each curve relative to the pure iterative baseline.
• Figure 4: Gauge-fixing performance for the RC48x48 ensemble across 100 test configurations. The plot compares the pure iterative baseline (blue, starting with 300 LA steps) against the hybrid approach with trained parameters from L21S2-1N-Z scheme (orange). The panels illustrate: (Top) The evolution of the gauge-fixing functional $F[g]$, where solid lines represent mean values and shaded bands indicate standard deviations. (Middle) The relative difference $\Delta F[g]$ between successive iterations; the bands capture the range between the maximum and minimum values among incomplete configurations. (Bottom) The count of remaining incomplete configurations as a function of SD iterations. As indicated in the legend, the hybrid approach achieves a normalized total computational cost of 0.9753 relative to the pure iterative baseline.