Flow-based sampling for multimodal and extended-mode distributions in lattice field theory

TL;DR

This work tackles the difficulty of sampling lattice field theories with multimodal and extended-mode distributions by advancing flow-based methods. It presents architecture- and training-based strategies—equivariant flows, topology matching, mixture models, forwards KL training, adiabatic retraining, and flow-distance regularization—to mitigate mode collapse and topological sampling issues. Through detailed experiments on real and complex 2D scalar field theories, the authors show significant improvements over baselines, and demonstrate that combining flow-based proposals with intermittent HMC updates yields robust, efficient sampling. The results illuminate both the promise and the limitations of flow-based inference in nonperturbative QFT, highlighting tail-region modeling as a critical area for further method development and optimization.

Abstract

Recent results have demonstrated that samplers constructed with flow-based generative models are a promising new approach for configuration generation in lattice field theory. In this paper, we present a set of training- and architecture-based methods to construct flow models for targets with multiple separated modes (i.e.~vacua) as well as targets with extended/continuous modes. We demonstrate the application of these methods to modeling two-dimensional real and complex scalar field theories in their symmetry-broken phases. In this context we investigate different flow-based sampling algorithms, including a composite sampling algorithm where flow-based proposals are occasionally augmented by applying updates using traditional algorithms like HMC.

Figures (39)

• Figure 1: Distribution of the average magnetization $\overline{\phi}$ in real scalar field theory for \ref{['fig:hmc-hists-M2']} fixed $\alpha=0$ varying $m^2$ and \ref{['fig:hmc-hists-alpha']} fixed $m^2=-4$ varying $\alpha$, at $\lambda=1$ on a $10 \times 10$ lattice geometry. Each is histogram estimated using $1.28 \times 10^6$ samples generated with AHMC.
• Figure 2: Reverse KL training history exhibiting smooth approach from unimodal to bimodal, for a flow model targeting real scalar field theory with $m^2=-3.6$ and $\alpha=0$, trained with batch size 16000. The ESS is computed every 25 epochs.
• Figure 3: Reverse KL training history exhibiting mode collapse, for a flow model targeting real scalar field theory with $m^2=-5$ and $\alpha=0$, trained with batch size 16000. The ESS is computed every 25 epochs. The similar values of $\text{ESS}/N$ and $\langle \overline{\phi} \rangle_{\tilde{p}}$ are coincidental.
• Figure 4: Reverse KL training history exhibiting tunneling from unimodal to bimodal, for a flow model targeting real scalar field theory with ${m^2=-4}$ and $\alpha=0$, training with batch size 16000 and using gradient norm clipping. The ESS is computed every 25 epochs.
• Figure 5: For real scalar field theory with $m^2=-4$ and $\alpha=0$, distribution of $\overline{\phi}$ for example models constructed using the different architectural approaches described in the text: equivariance with symmetrization (Symm.) or canonicalization (Canon.), and topology matching using a bimodal prior distribution (Topo.-matched). All models were trained for 10000 epochs. Note: these histograms are computed using raw model samples, and no statistical corrections have been applied. The target distribution ($p$), as measured using AHMC, is shown for comparison. All histograms are computed with $1.28 \times 10^6$ samples.