Particle Monte Carlo methods for Lattice Field Theory

TL;DR

This work addresses the challenge of sampling high-dimensional, multimodal distributions in lattice field theory (LFT) by evaluating GPU-accelerated particle Monte Carlo methods. It shows that simple, black-box sequential Monte Carlo and nested sampling approaches, tuned with a single data-driven covariance, can match or exceed state-of-the-art neural samplers on a canonical $\phi^4$ lattice benchmark in $1{+}1$D, while also enabling partition function estimation via $\log Z$. The key contributions are the systematic comparison of four particle baselines (SMC–RW, SMC–IRMH, SMC–HMC, NS) against neural baselines and a physics-informed AHMC reference, demonstrating strong performance across sample quality metrics (MMD, $W_2$) and scalability to larger lattices. The findings suggest that classical, accelerator-friendly methods can serve as robust baselines, potentially enabling hybrid strategies that combine particle methods with learned proposals to balance cost and accuracy in LFT and related high-dimensional inference tasks.

Abstract

High-dimensional multimodal sampling problems from lattice field theory (LFT) have become important benchmarks for machine learning assisted sampling methods. We show that GPU-accelerated particle methods, Sequential Monte Carlo (SMC) and nested sampling, provide a strong classical baseline that matches or outperforms state-of-the-art neural samplers in sample quality and wall-clock time on standard scalar field theory benchmarks, while also estimating the partition function. Using only a single data-driven covariance for tuning, these methods achieve competitive performance without problem-specific structure, raising the bar for when learned proposals justify their training cost.

Figures (6)

• Figure 1: Posterior magnetization density histograms for AHMC (reference), NS, SMC-RW, SMC-HMC, and SMC-IRMH.
• Figure 2: Eight $10\times 10$ field configurations sampled by NS at $\beta=1.0$ (top) and $\beta=0.0$ (bottom). Colors indicate site-wise fields $\phi_i$ with a shared colorbar; each panel is annotated with its mean magnetization $\langle\phi\rangle$.
• Figure 3: Magnetization across inverse temperature $\beta$: (a) tempered histograms of $\langle\phi\rangle$ at five selected $\beta$ values; (b) density over $(\beta,\langle\phi\rangle)$ across the full temperature range.
• Figure 4: MCMC kernel acceptance rate during SMC annealing. This diagnostic plot shows the average acceptance rate of the internal MCMC kernel at each step of the annealing process for SMC-RW, SMC-IRMH, and SMC-HMC. This metric reveals the efficiency of the proposal mechanism as the algorithm transitions from the prior to the posterior.
• Figure 5: Posterior magnetization density as a function of increasing lattice size. Each panel compares the ability of the sampling methods to capture the bimodal posterior distribution. As the lattice dimension grows, the two modes become more sharply peaked and separated, increasing the difficulty of the sampling problem. Quantitative details relating to this scaling are listed in \ref{['tab:scaling_analysis_full']}.