Scaling of Stochastic Normalizing Flows in $\mathrm{SU}(3)$ lattice gauge theory

TL;DR

This work demonstrates the first implementation of Stochastic Normalizing Flows for SU(3) lattice gauge theory in four dimensions and provides evidence that SNFs scale with the system's degrees of freedom in the same way as NE-MCMC. By interleaving gauge-equivariant NF layers with out-of-equilibrium MC updates, the authors achieve roughly a twofold improvement in key sampling metrics like the KL divergence and the Effective Sample Size, while keeping training costs modest. The study shows that the performance improvements persist across volumes and lattice spacings, with scaling governed by the ratio $n_{\mathrm{step}}/(L/a)^4$. These results point to SNFs as a scalable and practical approach to mitigate critical slowing down in high-dimensional gauge theories and motivate future work on protocol optimization and more expressive equivariant layers.

Abstract

Non-equilibrium Markov Chain Monte Carlo (NE-MCMC) simulations provide a well-understood framework based on Jarzynski's equality to sample from a target probability distribution. By driving a base probability distribution out of equilibrium, observables are computed without the need to thermalize. If the base distribution is characterized by mild autocorrelations, this approach provides a way to mitigate critical slowing down. Out-of-equilibrium evolutions share the same framework of flow-based approaches and they can be naturally combined into a novel architecture called Stochastic Normalizing Flows (SNFs). In this work we present the first implementation of SNFs for $\mathrm{SU}(3)$ lattice gauge theory in 4 dimensions, defined by introducing gauge-equivariant layers between out-of-equilibrium Monte Carlo updates. The core of our analysis is focused on the promising scaling properties of this architecture with the degrees of freedom of the system, which are directly inherited from NE-MCMC. Finally, we discuss how systematic improvements of this approach can realistically lead to a general and yet efficient sampling strategy at fine lattice spacings for observables affected by long autocorrelation times.

Figures (7)

• Figure 1: Value of the learned parameter $\rho^{(n)}$ along the SNF for the $n$-th layer (top panel) and the same value rescaled for $n_{\mathrm{step}}$ (bottom panel), for a training performed on a $L/a=12$ lattice for three values of $n_{\mathrm{step}}$ for a flow between $\beta=6.02$ and $\beta=6.178$.
• Figure 2: Results for the KL divergence (top panel) and for the ESS (bottom panel) for a flow between $\beta=6.02$ and $\beta=6.178$, for $L/a=12$ and $L/a=16$, for NE-MCMC (diamonds) and SNF (circles) architectures.
• Figure 3: Results for the KL divergence (top panel) and for the ESS (bottom panel) for all the lattice sizes analyzed in this study for a flow between $\beta=6.02$ and $\beta=6.178$, plotted against $n_{\mathrm{step}}/(L/a)^4$, for NE-MCMC (diamonds) and SNF (circles) architectures. $L/a=12$ data are connected with a dashed line to guide the eye.
• Figure 4: Results for the KL divergence (top panel) and for the ESS (bottom panel) for all the lattice sizes analyzed in this study for a flow between $\beta=5.896$ and $\beta=6.037$, plotted against $n_{\mathrm{step}}/(L/a)^4$, for NE-MCMC (diamonds) and SNF (circles) architectures. $L/a=12$ data are connected with a dashed line to guide the eye.
• Figure 5: Results for the KL divergence for $L/a=12$ for flows starting from $\beta=6.0$, with linear protocol and different increases $\delta \beta = (\beta - \beta_0)/n_{\mathrm{step}}$ at each step. Results are plotted against the inverse coupling (top panel) or against the change in inverse coupling normalized by $\delta \beta$ (bottom panel).