Variational Autoregressive Networks Applied to $\phi^4$ Field Theory Systems

We combine 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. 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 $\kappa$ 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 $\phi^4$ theory, our results agree with standard Monte Carlo benchmarks within errors, and no clear critical slowing down is observed in the explored parameter ranges.