# Learning the Ising Model with Generative Neural Networks

arXiv: https://arxiv.org/abs/2001.05361 · HTML: https://mlsanity.com/paper/2001.05361/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jan 2020.

## Built on by (most cited first, 9 of 9)

- [Multilevel Generative Samplers for Investigating Critical Phenomena](https://mlsanity.com/paper/2503.08918/) (2025)
- [Fine-tuning Vision Transformers for the Prediction of State Variables in Ising Models](https://mlsanity.com/paper/2109.13925/) (2021)
- [Approximate sampling and estimation of partition functions using neural networks](https://mlsanity.com/paper/2209.10423/) (2022)
- [Can a CNN trained on the Ising model detect the phase transition of the q-state Potts model?](https://mlsanity.com/paper/2104.03632/) (2021)
- [Discriminating modelling approaches for Point in Time Economic Scenario Generation](https://mlsanity.com/paper/2108.08818/) (2021)
- [Physics-Guided Problem Decomposition for Scaling Deep Learning of High-dimensional Eigen-Solvers: The Case of Schr"odinger's Equation](https://mlsanity.com/paper/2202.05994/) (2022)
- [Statistical Mechanics and Artificial Neural Networks: Principles, Models, and Applications](https://mlsanity.com/paper/2405.10957/) (2024)
- [On the flow matching interpretability](https://mlsanity.com/paper/2510.21210/) (2025)
- [Thermodynamic properties of chemically disordered compounds via AI-driven estimation of partition function with the PULSE method](https://mlsanity.com/paper/2605.28594/) (2026)

## Abstract (arXiv)

Recent advances in deep learning and neural networks have led to an increased interest in the application of generative models in statistical and condensed matter physics. In particular, restricted Boltzmann machines (RBMs) and variational autoencoders (VAEs) as specific classes of neural networks have been successfully applied in the context of physical feature extraction and representation learning. Despite these successes, however, there is only limited understanding of their representational properties and limitations. To better understand the representational characteristics of RBMs and VAEs, we study their ability to capture physical features of the Ising model at different temperatures. This approach allows us to quantitatively assess learned representations by comparing sample features with corresponding theoretical predictions. Our results suggest that the considered RBMs and convolutional VAEs are able to capture the temperature dependence of magnetization, energy, and spin-spin correlations. The samples generated by RBMs are more evenly distributed across temperature than those generated by VAEs. We also find that convolutional layers in VAEs are important to model spin correlations whereas RBMs achieve similar or even better performances without convolutional filters.