# Equivariant Symmetry Breaking Sets

arXiv: https://arxiv.org/abs/2402.02681 · HTML: https://mlsanity.com/paper/2402.02681/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 55 references that are in the map.
- First posted on arXiv 5 Feb 2024.

## Builds on (most cited first, 26 of 26)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Object-Centric Learning with Slot Attention](https://mlsanity.com/paper/2006.15055/) (2020)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Graph Normalizing Flows](https://mlsanity.com/paper/1905.13177/) (2019)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [Independent SE(3)-Equivariant Models for End-to-End Rigid Protein Docking](https://mlsanity.com/paper/2111.07786/) (2021)
- [Relaxing Equivariance Constraints with Non-stationary Continuous Filters](https://mlsanity.com/paper/2204.07178/) (2022)
- [The Surprising Effectiveness of Equivariant Models in Domains with Latent Symmetry](https://mlsanity.com/paper/2211.09231/) (2022)
- [Finding Symmetry Breaking Order Parameters with Euclidean Neural Networks](https://mlsanity.com/paper/2007.02005/) (2020)
- [Symmetry Breaking and Equivariant Neural Networks](https://mlsanity.com/paper/2312.09016/) (2023)
- [Improved machine learning algorithm for predicting ground state properties](https://mlsanity.com/paper/2301.13169/) (2023)
- [Symphony: Symmetry-Equivariant Point-Centered Spherical Harmonics for 3D Molecule Generation](https://mlsanity.com/paper/2311.16199/) (2023)
- [Approximately Equivariant Graph Networks](https://mlsanity.com/paper/2308.10436/) (2023)
- [Breaking the Symmetry: Resolving Symmetry Ambiguities in Equivariant Neural Networks](https://mlsanity.com/paper/2210.16646/) (2022)

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

- [Improving Equivariant Networks with Probabilistic Symmetry Breaking](https://mlsanity.com/paper/2503.21985/) (2025)
- [Latent Mixture of Symmetries for Sample-Efficient Dynamic Learning](https://mlsanity.com/paper/2510.03578/) (2025)
- [R2Det: Exploring Relaxed Rotation Equivariance in 2D object detection](https://mlsanity.com/paper/2408.11760/) (2024)
- [Equivariant Flow Matching for Symmetry-Breaking Bifurcation Problems](https://mlsanity.com/paper/2509.03340/) (2025)
- [When GNNs meet symmetry in ILPs: an orbit-based feature augmentation approach](https://mlsanity.com/paper/2501.14211/) (2025)
- [Transferable Learning of Reaction Pathways from Geometric Priors](https://mlsanity.com/paper/2504.15370/) (2025)
- [Any-Subgroup Equivariant Networks via Symmetry Breaking](https://mlsanity.com/paper/2603.19486/) (2026)
- [Energy Loss Functions for Physical Systems](https://mlsanity.com/paper/2511.02087/) (2025)
- [Terminal Symmetry as a Carrier of Asymmetric Process Knowledge: Statewise Refinement for Anytime Verified Construction](https://mlsanity.com/paper/2608.11318/) (2026)

## Abstract (arXiv)

Equivariant neural networks (ENNs) have been shown to be extremely effective in applications involving underlying symmetries. By construction ENNs cannot produce lower symmetry outputs given a higher symmetry input. However, symmetry breaking occurs in many physical systems and we may obtain a less symmetric stable state from an initial highly symmetric one. Hence, it is imperative that we understand how to systematically break symmetry in ENNs. In this work, we propose a novel symmetry breaking framework that is fully equivariant and is the first which fully addresses spontaneous symmetry breaking. We emphasize that our approach is general and applicable to equivariance under any group. To achieve this, we introduce the idea of symmetry breaking sets (SBS). Rather than redesign existing networks, we design sets of symmetry breaking objects which we feed into our network based on the symmetry of our inputs and outputs. We show there is a natural way to define equivariance on these sets, which gives an additional constraint. Minimizing the size of these sets equates to data efficiency. We prove that minimizing these sets translates to a well studied group theory problem, and tabulate solutions to this problem for the point groups. Finally, we provide some examples of symmetry breaking to demonstrate how our approach works in practice. The code for these examples is available at urlhttps://github.com/atomicarchitects/equivariant-SBS.