# Simulating first-order phase transition with hierarchical autoregressive networks

arXiv: https://arxiv.org/abs/2212.04955 · HTML: https://mlsanity.com/paper/2212.04955/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 30 references that are in the map.
- First posted on arXiv 9 Dec 2022.

## Builds on (most cited first, 10 of 10)

- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Solving Statistical Mechanics Using Variational Autoregressive Networks](https://mlsanity.com/paper/1809.10606/) (2018)
- [Introduction to Normalizing Flows for Lattice Field Theory](https://mlsanity.com/paper/2101.08176/) (2021)
- [Gauge-equivariant flow models for sampling in lattice field theories with pseudofermions](https://mlsanity.com/paper/2207.08945/) (2022)
- [Flow-based sampling for fermionic lattice field theories](https://mlsanity.com/paper/2106.05934/) (2021)
- [Flow-based sampling for multimodal and extended-mode distributions in lattice field theory](https://mlsanity.com/paper/2107.00734/) (2021)
- [Flow-based sampling in the lattice Schwinger model at criticality](https://mlsanity.com/paper/2202.11712/) (2022)
- [Hierarchical autoregressive neural networks for statistical systems](https://mlsanity.com/paper/2203.10989/) (2022)
- [Analysis of autocorrelation times in Neural Markov Chain Monte Carlo simulations](https://mlsanity.com/paper/2111.10189/) (2021)
- [Machine Learning Trivializing Maps: A First Step Towards Understanding How Flow-Based Samplers Scale Up](https://mlsanity.com/paper/2112.15532/) (2021)

## Built on by (most cited first, 5 of 5)

- [Mutual information of spin systems from autoregressive neural networks](https://mlsanity.com/paper/2304.13412/) (2023)
- [Training normalizing flows with computationally intensive target probability distributions](https://mlsanity.com/paper/2308.13294/) (2023)
- [Deep generative modelling of canonical ensemble with differentiable thermal properties](https://mlsanity.com/paper/2404.18404/) (2024)
- [Sampling two-dimensional spin systems with transformers](https://mlsanity.com/paper/2604.27738/) (2026)
- [Variational Autoregressive Networks with probability priors](https://mlsanity.com/paper/2605.16020/) (2026)

## Abstract (arXiv)

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 106 configurations.