# A Group-Equivariant Autoencoder for Identifying Spontaneously Broken Symmetries

arXiv: https://arxiv.org/abs/2202.06319 · HTML: https://mlsanity.com/paper/2202.06319/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 69 references that are in the map.
- First posted on arXiv 13 Feb 2022.

## Builds on (most cited first, 7 of 7)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Don't Decay the Learning Rate, Increase the Batch Size](https://mlsanity.com/paper/1711.00489/) (2017)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Discovering Phase Transitions with Unsupervised Learning](https://mlsanity.com/paper/1606.00318/) (2016)
- [Unsupervised learning of phase transitions: from principal component analysis to variational autoencoders](https://mlsanity.com/paper/1703.02435/) (2017)
- [Deep Learning the Ising Model Near Criticality](https://mlsanity.com/paper/1708.04622/) (2017)
- [Variational Autoencoder Analysis of Ising Model Statistical Distributions and Phase Transitions](https://mlsanity.com/paper/2104.06368/) (2021)

## Built on by (most cited first, 2 of 2)

- [Interpretable Machine Learning in Physics: A Review](https://mlsanity.com/paper/2503.23616/) (2025)
- [Densely Connected G-invariant Deep Neural Networks with Signed Permutation Representations](https://mlsanity.com/paper/2303.04614/) (2023)

## Abstract (arXiv)

We introduce the group-equivariant autoencoder (GE-autoencoder) -- a deep neural network (DNN) method that locates phase boundaries by determining which symmetries of the Hamiltonian have spontaneously broken at each temperature. We use group theory to deduce which symmetries of the system remain intact in all phases, and then use this information to constrain the parameters of the GE-autoencoder such that the encoder learns an order parameter invariant to these ``never-broken'' symmetries. This procedure produces a dramatic reduction in the number of free parameters such that the GE-autoencoder size is independent of the system size. We include symmetry regularization terms in the loss function of the GE-autoencoder so that the learned order parameter is also equivariant to the remaining symmetries of the system. By examining the group representation by which the learned order parameter transforms, we are then able to extract information about the associated spontaneous symmetry breaking. We test the GE-autoencoder on the 2D classical ferromagnetic and antiferromagnetic Ising models, finding that the GE-autoencoder (1) accurately determines which symmetries have spontaneously broken at each temperature; (2) estimates the critical temperature in the thermodynamic limit with greater accuracy, robustness, and time-efficiency than a symmetry-agnostic baseline autoencoder; and (3) detects the presence of an external symmetry-breaking magnetic field with greater sensitivity than the baseline method. Finally, we describe various key implementation details, including a new method for extracting the critical temperature estimate from trained autoencoders and calculations of the DNN initialization and learning rate settings required for fair model comparisons.