Variational Autoregressive Networks Applied to $φ^4$ Field Theory Systems

TL;DR

This work combines reinforcement learning with variational autoregressive networks (VANs) to perform data-free training and sampling for the discrete Ising model and the continuous $\phi^4$ scalar field theory, and introduces single-site and block Metropolis--Hastings updates on top of VAN proposals.

Abstract

We combine reinforcement learning with variational autoregressive networks (VANs) to perform data-free training and sampling for the discrete Ising model and the continuous $φ^4$ scalar field theory. We quantify the complexity of the target distribution via the KL divergence between the magnetization distribution and a reference Gaussian distribution, and observe that configurations with smaller KL divergence typically require fewer training steps. Motivated by this observation, we investigate transfer learning and show that fine-tuning models pretrained at a single value of $κ$ can reduce training time compared with training from a Gaussian field. In addition, inspired by single-site and cluster Monte Carlo updates, we introduce single-site and block Metropolis--Hastings (MH) updates on top of VAN proposals. These MH corrections systematically reduce the residual bias of pure VAN sampling in the parameter range we study, while maintaining high sampling efficiency in terms of the effective sample size (ESS). For both the Ising model and the $φ^4$ theory, our results agree with standard Monte Carlo benchmarks within errors, and no clear critical slowing down is observed in the explored parameter ranges.

Figures (13)

• Figure 1: Illustration of the VAN sampling process on a two-dimensional Gaussian mixture. Left: prior distribution $p_0 = \mathcal{N}(0,1)$. Middle: sampling trajectories---step 1 draws $s_1$ from $p(s_1)$ (horizontal moves), step 2 draws $s_2$ from $p(s_2|s_1)$ (vertical moves). Blue and red trajectories are routed to the left and right modes, respectively. Right: the target bimodal distribution $p(s_1, s_2)$.
• Figure 2: Autoregressive sampling process of a VAN on a $3\times 3$ Ising model. Top row: sequential evolution of lattice configurations, with the current sampling site highlighted by an orange frame. Middle row: conditional distributions $p_\theta(\sigma_i|\sigma_{<i})$ output by the network at each step, with the sampled value highlighted by an orange frame. Bottom row: sampling descriptions, showing the conditional formulas, sampled spins, and partial energy contributions. The network learns nontrivial patterns: for example, in step 2, despite ferromagnetic nearest-neighbor couplings, one finds $P(\sigma_{01}=-1|\sigma_{00}=+1) = 0.75$, reflecting global energy optimization.
• Figure 3: Thermodynamic observables for the Ising model ($L=6$).(a):Magnetization;(b):Susceptibility. Comparison between VAN (red circles), VAN+MH (green circles), and MC benchmarks (black triangles).
• Figure 4: Effective sample size (ESS) as a function of inverse temperature $\beta$ for the Ising model ($L=6$). ESS remains uniformly high across all temperatures, indicating efficient sampling. Results with and without transfer learning overlap, indicating that transfer learning does not significantly affect ESS.
• Figure 5: Thermodynamic observables of the $\phi^4$ theory ($L=6$). Blue: pure VAN sampling; orange: VAN with single-site MH; black: HMC benchmarks.