We apply the Hierarchical Autoregressive Neural (HAN) network sampling algorithm to the two-dimensional $Q$-state Potts model and perform simulations around the phase transition at $Q=12$. We quantify the performance of the approach in the vicinity of the first-order phase transition and compare it with that of the Wolff cluster algorithm. We find a significant improvement as far as the statistical uncertainty is concerned at a similar numerical effort. In order to efficiently train large neural networks we introduce the technique of pre-training. It allows to train some neural networks using smaller system sizes and then employing them as starting configurations for larger system sizes. This is possible due to the recursive construction of our hierarchical approach. Our results serve as a demonstration of the performance of the hierarchical approach for systems exhibiting bimodal distributions. Additionally, we provide estimates of the free energy and entropy in the vicinity of the phase transition with statistical uncertainties of the order of $10^{-7}$ for the former and $10^{-3}$ for the latter based on a statistics of $10^6$ configurations.

翻译：我们应用层级自回归神经网络（HAN）采样算法于二维$Q$态Potts模型，并在$Q=12$处围绕相变进行模拟。我们量化了该方法在一级相变附近的性能表现，并与Wolff簇算法进行了比较。研究发现，在相近计算量下，统计不确定度得到显著改善。为高效训练大型神经网络，我们引入了预训练技术。该技术可利用较小系统尺寸训练部分神经网络，并将其作为较大系统尺寸的初始配置。这得益于我们层级方法中的递归构造。我们的结果证明了层级方法在具有双峰分布系统中的性能表现。此外，我们基于$10^6$个配置的统计数据，给出了相变附近自由能和熵的估计值，其统计不确定度分别达到$10^{-7}$量级和$10^{-3}$量级。