Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Group-Equivariant Diffusion Models for Lattice Field Theory

Octavio Vega, Javad Komijani, Aida El-Khadra, Marina Marinkovic

TL;DR

This work tackles critical slowing down in lattice quantum field theory sampling by introducing symmetry-preserving, score-based diffusion models. It develops group-equivariant score networks for transformations including $\mathbb{Z}_2$, ${\rm U}(1)$, and translations, and introduces force-regularized score matching to inject physical priors, enabling exact likelihood-based sampling. The authors demonstrate the approach on 2D scalar $\phi^4$ theory and 2D ${\rm U}(1)$ lattice gauge theory, achieving high effective sample sizes and strong agreement with standard MCMC observables, while substantially reducing autocorrelations. The results indicate that enforcing exact symmetries and physics-informed training improves expressivity and sampling efficiency, with promising potential for extension to non-Abelian gauge theories and fermionic degrees of freedom.

Abstract

Near the critical point, Markov Chain Monte Carlo (MCMC) simulations of lattice quantum field theories (LQFT) become increasingly inefficient due to critical slowing down. In this work, we investigate score-based symmetry-preserving diffusion models as an alternative strategy to sample two-dimensional $φ^4$ and ${\rm U}(1)$ lattice field theories. We develop score networks that are equivariant to a range of group transformations, including global $\mathbb{Z}_2$ reflections, local ${\rm U}(1)$ rotations, and periodic translations $\mathbb{T}$. The score networks are trained using an augmented training scheme, which significantly improves sample quality in the simulated field theories. We also demonstrate empirically that our symmetry-aware models outperform generic score networks in sample quality, expressivity, and effective sample size.

Group-Equivariant Diffusion Models for Lattice Field Theory

TL;DR

This work tackles critical slowing down in lattice quantum field theory sampling by introducing symmetry-preserving, score-based diffusion models. It develops group-equivariant score networks for transformations including , , and translations, and introduces force-regularized score matching to inject physical priors, enabling exact likelihood-based sampling. The authors demonstrate the approach on 2D scalar theory and 2D lattice gauge theory, achieving high effective sample sizes and strong agreement with standard MCMC observables, while substantially reducing autocorrelations. The results indicate that enforcing exact symmetries and physics-informed training improves expressivity and sampling efficiency, with promising potential for extension to non-Abelian gauge theories and fermionic degrees of freedom.

Abstract

Near the critical point, Markov Chain Monte Carlo (MCMC) simulations of lattice quantum field theories (LQFT) become increasingly inefficient due to critical slowing down. In this work, we investigate score-based symmetry-preserving diffusion models as an alternative strategy to sample two-dimensional and lattice field theories. We develop score networks that are equivariant to a range of group transformations, including global reflections, local rotations, and periodic translations . The score networks are trained using an augmented training scheme, which significantly improves sample quality in the simulated field theories. We also demonstrate empirically that our symmetry-aware models outperform generic score networks in sample quality, expressivity, and effective sample size.

Paper Structure

This paper contains 28 sections, 1 theorem, 104 equations, 12 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Building Diffusion Models for LQFT
  3. Diffusion as a Reversible Stochastic Process
  4. Symmetries of Score Functions
  5. Training through Force-Regularized Score Matching
  6. Likelihood Computations and Exactness
  7. Toy Example in 0D
  8. $\mathbb{Z}_2$ Symmetry
  9. Training and Inference
  10. Effective Sample Sizes
  11. Application to Scalar $\phi^4$ Theory in 2D
  12. Network Architecture
  13. Numerical Experiments
  14. Influence of Symmetry on Model Expressivity
  15. Application to ${\rm U}(1)$ Gauge Theory in 2D
  16. ...and 13 more sections

Key Result

Theorem C.4

Let $G \in \mathcal{G}$ act smoothly on $\mathcal{F}$ by $\phi \mapsto \phi' = \rho_G(\phi)$. If $p_t$ is $\mathcal{G}$-invariant, then the score function transforms under the group action as where the Jacobian (differential) $D_\phi \rho_G := \partial \rho_G(\phi) / \partial \phi$ represents the linearization of the group action at $\phi$.

Figures (12)

  • Figure 1: Visualization of the forward diffusion process over time in the variance-preserving picture. The histogram (blue) of data samples is seen to evolve into a Gaussian distribution while the variance is approximately constant in time. The red lines represent three sample trajectories of the data stochastically evolving towards $t=1$.
  • Figure 2: Learned scores as functions of the field variable $\phi$ in the (left) symmetric and (right) broken phases. The blue and red solid lines correspond to the score functions near the start ($\tau = 0.01$) and end ($\tau = 0.99$) of the reverse process, respectively. The dotted black line represents the true force, i.e. $-\nabla_\phi S(\phi)$.
  • Figure 3: Normalized histograms with 80 bins over 16,384 samples of training data (HMC) and 2,048 samples of diffusion model (DM) generated data for (left) the symmetric and (right) broken phases. The coupling strength was set to $\lambda = 0.4$ for both phases, and the squared mass is $m^2 = 1.0$ and $m^2 = -1.0$ for the two phases, respectively.
  • Figure 4: Evolution of the effective action in the broken phase as a function of the field $\phi$ over reverse diffusion time. The colorbar is parameterized by (forward) diffusion time $t \equiv 1 -\tau$.
  • Figure 5: Evolution of (left) the effective action $S_{\rm eff}(\phi, t)$ and (right) the resulting density $p(\phi, t) \propto e^{-S_{\rm eff}(\phi, t)}$ over reverse diffusion time $\tau$ as a function of the field $\phi$.
  • ...and 7 more figures

Theorems & Definitions (5)

  • Definition C.1: Invariance
  • Definition C.2: Equivariance
  • Definition C.3: Score Function
  • Theorem C.4: Transformation Law of the Score Function
  • proof