# Finding Symmetry Breaking Order Parameters with Euclidean Neural Networks

arXiv: https://arxiv.org/abs/2007.02005 · HTML: https://mlsanity.com/paper/2007.02005/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jul 2020.

## Built on by (most cited first, 30 of 30)

- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Geometric Deep Learning on Molecular Representations](https://mlsanity.com/paper/2107.12375/) (2021)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Geometric Clifford Algebra Networks](https://mlsanity.com/paper/2302.06594/) (2023)
- [Applications and Techniques for Fast Machine Learning in Science](https://mlsanity.com/paper/2110.13041/) (2021)
- [Symmetry Breaking and Equivariant Neural Networks](https://mlsanity.com/paper/2312.09016/) (2023)
- [Equivariant Networks for Crystal Structures](https://mlsanity.com/paper/2211.15420/) (2022)
- [Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolution](https://mlsanity.com/paper/2310.02299/) (2023)
- [Equivariance via Minimal Frame Averaging for More Symmetries and Efficiency](https://mlsanity.com/paper/2406.07598/) (2024)
- [Equivariant Filters for Efficient Tracking in 3D Imaging](https://mlsanity.com/paper/2103.10255/) (2021)
- [Improving Equivariant Networks with Probabilistic Symmetry Breaking](https://mlsanity.com/paper/2503.21985/) (2025)
- [E2Former: An Efficient and Equivariant Transformer with Linear-Scaling Tensor Products](https://mlsanity.com/paper/2501.19216/) (2025)
- [Towards out of distribution generalization for problems in mechanics](https://mlsanity.com/paper/2206.14917/) (2022)
- [ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitals](https://mlsanity.com/paper/2503.08305/) (2025)
- [Learning (Approximately) Equivariant Networks via Constrained Optimization](https://mlsanity.com/paper/2505.13631/) (2025)
- [Equivariant Symmetry Breaking Sets](https://mlsanity.com/paper/2402.02681/) (2024)
- [Equivariant Denoisers Cannot Copy Graphs: Align Your Graph Diffusion Models](https://mlsanity.com/paper/2405.17656/) (2024)
- [Relaxed Equivariant Graph Neural Networks](https://mlsanity.com/paper/2407.20471/) (2024)
- [R2Det: Exploring Relaxed Rotation Equivariance in 2D object detection](https://mlsanity.com/paper/2408.11760/) (2024)
- [When GNNs meet symmetry in ILPs: an orbit-based feature augmentation approach](https://mlsanity.com/paper/2501.14211/) (2025)
- [Deconstructing equivariant representations in molecular systems](https://mlsanity.com/paper/2410.08131/) (2024)
- [Any-Subgroup Equivariant Networks via Symmetry Breaking](https://mlsanity.com/paper/2603.19486/) (2026)
- [Spontaneous symmetry breaking and Goldstone modes for deep information propagation](https://mlsanity.com/paper/2605.14685/) (2026)
- [U(1) Symmetry-breaking Observed in Generic CNN Bottleneck Layers](https://mlsanity.com/paper/2206.02220/) (2022)
- [Energy Loss Functions for Physical Systems](https://mlsanity.com/paper/2511.02087/) (2025)
- [Cross-Chirality Generalization by Axial Vectors for Hetero-Chiral Protein-Peptide Interaction Design](https://mlsanity.com/paper/2602.20176/) (2026)
- [Automorphism-Induced Non-Canonicity in Top-k Explanations of Graph Neural Networks](https://mlsanity.com/paper/2607.26344/) (2026)
- [Reducing Symmetry Increase in Equivariant Neural Networks](https://mlsanity.com/paper/2608.12010/) (2026)
- [Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs](https://mlsanity.com/paper/2608.28853/) (2026)

## Abstract (arXiv)

Curie's principle states that "when effects show certain asymmetry, this asymmetry must be found in the causes that gave rise to them". We demonstrate that symmetry equivariant neural networks uphold Curie's principle and can be used to articulate many symmetry-relevant scientific questions into simple optimization problems. We prove these properties mathematically and demonstrate them numerically by training a Euclidean symmetry equivariant neural network to learn symmetry-breaking input to deform a square into a rectangle and to generate octahedra tilting patterns in perovskites.