# Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolution

arXiv: https://arxiv.org/abs/2310.02299 · HTML: https://mlsanity.com/paper/2310.02299/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 71 references that are in the map.
- First posted on arXiv 3 Oct 2023.

## Builds on (most cited first, 30 of 42)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [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)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representations](https://mlsanity.com/paper/2306.12059/) (2023)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Spherical Message Passing for 3D Graph Networks](https://mlsanity.com/paper/2102.05013/) (2021)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Deep Learning for Time Series Forecasting: Tutorial and Literature Survey](https://mlsanity.com/paper/2004.10240/) (2020)
- [Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNs](https://mlsanity.com/paper/2302.03655/) (2023)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [Exploiting Cyclic Symmetry in Convolutional Neural Networks](https://mlsanity.com/paper/1602.02660/) (2016)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [Meta-Learning Symmetries by Reparameterization](https://mlsanity.com/paper/2007.02933/) (2020)
- [Residual Pathway Priors for Soft Equivariance Constraints](https://mlsanity.com/paper/2112.01388/) (2021)
- [GemNet-OC: Developing Graph Neural Networks for Large and Diverse Molecular Simulation Datasets](https://mlsanity.com/paper/2204.02782/) (2022)
- [Spherical Channels for Modeling Atomic Interactions](https://mlsanity.com/paper/2206.14331/) (2022)
- [Approximation-Generalization Trade-offs under (Approximate) Group Equivariance](https://mlsanity.com/paper/2305.17592/) (2023)

## Built on by (most cited first, 20 of 20)

- [Improving Equivariant Model Training via Constraint Relaxation](https://mlsanity.com/paper/2408.13242/) (2024)
- [Improving Equivariant Networks with Probabilistic Symmetry Breaking](https://mlsanity.com/paper/2503.21985/) (2025)
- [Learning (Approximately) Equivariant Networks via Constrained Optimization](https://mlsanity.com/paper/2505.13631/) (2025)
- [Approximate Equivariance in Reinforcement Learning](https://mlsanity.com/paper/2411.04225/) (2024)
- [A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Models](https://mlsanity.com/paper/2506.02269/) (2025)
- [Relaxed Equivariant Graph Neural Networks](https://mlsanity.com/paper/2407.20471/) (2024)
- [Achieving Approximate Symmetry Is Exponentially Easier than Exact Symmetry](https://mlsanity.com/paper/2512.11855/) (2025)
- [Learning Symmetries via Weight-Sharing with Doubly Stochastic Tensors](https://mlsanity.com/paper/2412.04594/) (2024)
- [Latent Mixture of Symmetries for Sample-Efficient Dynamic Learning](https://mlsanity.com/paper/2510.03578/) (2025)
- [Adaptive Canonicalization with Application to Invariant Anisotropic Geometric Networks](https://mlsanity.com/paper/2509.24886/) (2025)
- [MaNGO - Adaptable Graph Network Simulators via Meta-Learning](https://mlsanity.com/paper/2510.05874/) (2025)
- [Segment as You Wish -- Free-Form Language-Based Segmentation for Medical Images](https://mlsanity.com/paper/2410.12831/) (2024)
- [PEnGUiN: Partially Equivariant Graph NeUral Networks for Sample Efficient MARL](https://mlsanity.com/paper/2503.15615/) (2025)
- [Approximate Equivariance via Projection-based Regularisation](https://mlsanity.com/paper/2601.05028/) (2026)
- [Learning Ordering in Crystalline Materials with Symmetry-Aware Graph Neural Networks](https://mlsanity.com/paper/2409.13851/) (2024)
- [Exact Symmetry as Algebra: Group-Structured Tensors that Enforce Physical Selection Rules](https://mlsanity.com/paper/2605.20440/) (2026)
- [Implicit Augmentation from Distributional Symmetry in Turbulence Super-Resolution](https://mlsanity.com/paper/2509.20683/) (2025)
- [An Exploratory Study into using Machine-Learning for Fast Step-by-step Emulation of Numerical Mechanical Thrombectomy Simulations for Ischemic Stroke](https://mlsanity.com/paper/2606.00892/) (2026)
- [Terminal Symmetry as a Carrier of Asymmetric Process Knowledge: Statewise Refinement for Anytime Verified Construction](https://mlsanity.com/paper/2608.11318/) (2026)
- [Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs](https://mlsanity.com/paper/2608.28853/) (2026)

## Abstract (arXiv)

Modeling symmetry breaking is essential for understanding the fundamental changes in the behaviors and properties of physical systems, from microscopic particle interactions to macroscopic phenomena like fluid dynamics and cosmic structures. Thus, identifying sources of asymmetry is an important tool for understanding physical systems. In this paper, we focus on learning asymmetries of data using relaxed group convolutions. We provide both theoretical and empirical evidence that this flexible convolution technique allows the model to maintain the highest level of equivariance that is consistent with data and discover the subtle symmetry-breaking factors in various physical systems. We employ various relaxed group convolution architectures to uncover various symmetry-breaking factors that are interpretable and physically meaningful in different physical systems, including the phase transition of crystal structure, the isotropy and homogeneity breaking in turbulent flow, and the time-reversal symmetry breaking in pendulum systems.