Practical applications of machine-learned flows on gauge fields

TL;DR

The paper addresses the challenge of applying machine-learned flows to improve sampling of gauge fields in lattice QCD at scale. It introduces two flow-based replica-exchange strategies: Transformed Replica Exchange (T-REX), which uses a flow-bridge to boost swap acceptance between neighboring densities, and Defect Repair Replica Exchange (DR-REX), which applies flows to repair localized action defects and enhance mixing. On pure-gauge SU(3) lattices with the Wilson action, both methods improve topological mixing, with T-REX achieving swap rates of around 15-20% and reduced topological-charge autocorrelation times, while DR-REX yields meaningful swap rates for small defects. Although full computational advantages are not yet demonstrated once learned components are accounted for, the results reveal structural benefits and a clear path to scaling by larger defects and further flow development.

Abstract

Normalizing flows are machine-learned maps between different lattice theories which can be used as components in exact sampling and inference schemes. Ongoing work yields increasingly expressive flows on gauge fields, but it remains an open question how flows can improve lattice QCD at state-of-the-art scales. We discuss and demonstrate two applications of flows in replica exchange (parallel tempering) sampling, aimed at improving topological mixing, which are viable with iterative improvements upon presently available flows.

Figures (5)

• Figure 1: Evolution of the topological charge for a set of three different pure-gauge targets on a $12^4$ volume, sampled simultaneously using either REX or T-REX. REX swaps are almost never accepted, so it is equivalent to independent HB+OR on each stream. One MCMC step is 5 HB hits followed by 2 OR hits. Swaps are proposed every 5 steps.
• Figure 2: Running estimates of the integrated autocorrelation times of the topological charge for three pure-gauge actions, sampled simultaneously using either REX or T-REX as in \ref{['fig:rex-vs-trex-Q']}. Units are MCMC steps as in \ref{['fig:rex-vs-trex-Q']}. The curves show $1/2 + \sum_{\delta_t=1}^{t_\mathrm{max}} C_Q(\delta t)$ and are expected to plateau at $\tau_\mathrm{int}(Q)$. Uncertainties are evaluated as the standard error over 32 (40) independent streams for REX (T-REX), each of length 10000 (4500) steps. For T-REX, the $\beta=6.05$ curve is reproduced with a factor of 3 for ease of cost comparison.
• Figure 3: Sketches of the geometry for defect repair flows. At left, the flow performs the identity operation outside the orange region. The active region around the defect (orange) is acted upon, conditioned on its local context (blue). At right, the exact subvolume geometry used for the flows in this work. This geometry is symmetric in all spatial directions ($y$, $z$ not shown).
• Figure 4: Evolution of the topological charge under PTBC or DR-REX for pure-gauge targets with $\beta=6.3$ on a $16^4$ volume with OBC defects of size $2^3$ with $\beta_d=0,3,6.3$, where $\beta_d=\beta=6.3$ is the target stream with no defect. PTBC swaps are almost never accepted, so it is equivalent to independent HB+OR on each stream. One MCMC step is 1 HB hit followed by 5 OR hits. Swaps are proposed every 10 steps.
• Figure 5: Running estimates of the integrated autocorrelation times of the topological charge for sampling using PTBC or DR-REX as in \ref{['fig:ptbc-vs-drrex-Q']}. Units are MCMC steps as in \ref{['fig:ptbc-vs-drrex-Q']}. The curves show $1/2 + \sum_{\delta_t=1}^{t_\mathrm{max}} C_Q(\delta t)$ and are expected to plateau at $\tau_\mathrm{int}(Q)$. Uncertainties are evaluated as the standard error over 24 independent streams, each of length 29000 steps. For DR-REX, the $\beta_d=6.3$ curve is reproduced with a factor of 3 for ease of cost comparison.