Neural Field Transformations for Hybrid Monte Carlo: Architectural Design and Scaling

TL;DR

The paper addresses critical slowing down in Hybrid Monte Carlo sampling for lattice gauge theories and proposes neural field transformations within the NTHMC framework to reshape the energy landscape via an invertible, gauge-covariant map $\mathcal{F}$, producing an effective action $S_{\mathrm{FT}}(\tilde{U}) = S(\mathcal{F}(\tilde{U})) - \log| \det \mathcal{F}^*(\tilde{U}) |$ for sampling. It systematically analyzes architectural designs (e.g., residual connections, attention, channel-dependent activation) of gauge-covariant CNNs on a 2D $U(1)$ theory, training at $\beta=3$, $V=32^2$, and assessing transfer to larger volumes and finer lattices using autocorrelation $\gamma(\delta)$ and topological-change metrics $\Delta Q$. The results show that wider receptive fields and channel-dependent activations boost sampling efficiency, with the Comb architecture delivering the largest gains (approximately a 40% reduction in $\gamma(16)$) and improved topological tunneling, while maintaining favorable scaling across volume and lattice spacing. These findings provide concrete design guidelines for extending NTHMC to four-dimensional $SU(3)$ gauge theories and large-scale lattice QCD, with future work on integrator choices, training objectives beyond force minimization, and broader training regimes.

Abstract

Critical slowing down, where autocorrelation grows rapidly near the continuum limit due to Hybrid Monte Carlo (HMC) moving through configuration space inefficiently, still challenges lattice gauge theory simulations. Combining neural field transformations with HMC (NTHMC) can reshape the energy landscape and accelerate sampling, but the choice of neural architectures has yet to be studied systematically. We evaluate NTHMC on a two-dimensional U(1) gauge theory, analyzing how it scales and transfers to larger volumes and smaller lattice spacing. Controlled comparisons let us isolate architectural contributions to sampling efficiency. Good designs can reduce autocorrelation and boost topological tunneling while maintaining favorable scaling. More broadly, our study highlights emerging design guides, such as wider receptive fields and channel-dependent activations, paving the way for systematic extensions to four-dimensional SU(3) gauge theory.

Figures (3)

• Figure 1: Cropped section of a lattice configuration illustrating the field transformation. Red lines mark one of the eight link subsets updated sequentially, green lines indicate Wilson loops $W_{x,\mu,l}$, and blue lines show the gauge-invariant features $(X,Y,\dots)$ used as CNN inputs.
• Figure 2: Comparison of regular HMC and NTHMC with the Base model. Left: autocorrelation decay at $\beta=6.0$, $V=64^2$. Right: scaling of two efficiency ratios across $\beta$ and volume, with $R_{\gamma(16)} \equiv \gamma(16)_{\mathrm{HMC}} / \gamma(16)_{\mathrm{Base}}$ and $R_{\Delta Q} \equiv \Delta Q_{\mathrm{Base}} / \Delta Q_{\mathrm{HMC}}$.
• Figure 3: Comparison of NTHMC with various network architectures across different lattice couplings and volumes. The plots show $\Delta Q$ (left panel) and $\gamma(16)$ (right panel), both normalized to standard HMC results.