# Symmetry Group Equivariant Architectures for Physics

arXiv: https://arxiv.org/abs/2203.06153 · HTML: https://mlsanity.com/paper/2203.06153/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 49 references that are in the map.
- First posted on arXiv 11 Mar 2022.

## Builds on (most cited first, 29 of 29)

- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [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)
- [Network Dissection: Quantifying Interpretability of Deep Visual Representations](https://mlsanity.com/paper/1704.05796/) (2017)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (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)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Contrastive Learning of Structured World Models](https://mlsanity.com/paper/1911.12247/) (2019)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [LieTransformer: Equivariant self-attention for Lie Groups](https://mlsanity.com/paper/2012.10885/) (2020)
- [Exploiting Cyclic Symmetry in Convolutional Neural Networks](https://mlsanity.com/paper/1602.02660/) (2016)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [Residual Pathway Priors for Soft Equivariance Constraints](https://mlsanity.com/paper/2112.01388/) (2021)
- [N-body Networks: a Covariant Hierarchical Neural Network Architecture for Learning Atomic Potentials](https://mlsanity.com/paper/1803.01588/) (2018)
- [Deep learning in color: towards automated quark/gluon jet discrimination](https://mlsanity.com/paper/1612.01551/) (2016)
- [Group equivariant neural posterior estimation](https://mlsanity.com/paper/2111.13139/) (2021)
- [Semi-Equivariant GNN Architectures for Jet Tagging](https://mlsanity.com/paper/2202.06941/) (2022)

## Built on by (most cited first, 23 of 23)

- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [On the Expressive Power of Geometric Graph Neural Networks](https://mlsanity.com/paper/2301.09308/) (2023)
- [Group-Invariant Quantum Machine Learning](https://mlsanity.com/paper/2205.02261/) (2022)
- [Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.09974/) (2022)
- [Learning Probabilistic Symmetrization for Architecture Agnostic Equivariance](https://mlsanity.com/paper/2306.02866/) (2023)
- [Explainable Equivariant Neural Networks for Particle Physics: PELICAN](https://mlsanity.com/paper/2307.16506/) (2023)
- [Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles](https://mlsanity.com/paper/2301.05638/) (2023)
- [EdgeBench: Unveiling Scaling Laws of Learning from Real-World Environments](https://mlsanity.com/paper/2607.05155/) (2026)
- [Approximately Equivariant Graph Networks](https://mlsanity.com/paper/2308.10436/) (2023)
- [Improving Equivariant Networks with Probabilistic Symmetry Breaking](https://mlsanity.com/paper/2503.21985/) (2025)
- [Snowmass 2021 Computational Frontier CompF03 Topical Group Report: Machine Learning](https://mlsanity.com/paper/2209.07559/) (2022)
- [Symmetry-invariant quantum machine learning force fields](https://mlsanity.com/paper/2311.11362/) (2023)
- [Improved Canonicalization for Model Agnostic Equivariance](https://mlsanity.com/paper/2405.14089/) (2024)
- [Oracle-Preserving Latent Flows](https://mlsanity.com/paper/2302.00806/) (2023)
- [Equivariance Is Not All You Need: Characterizing the Utility of Equivariant Graph Neural Networks for Particle Physics Tasks](https://mlsanity.com/paper/2311.03094/) (2023)
- [Optimal Equivariant Architectures from the Symmetries of Matrix-Element Likelihoods](https://mlsanity.com/paper/2410.18553/) (2024)
- [From thermodynamics to protein design: Diffusion models for biomolecule generation towards autonomous protein engineering](https://mlsanity.com/paper/2501.02680/) (2025)
- [Stable and Interpretable Jet Physics with IRC-Safe Equivariant Feature Extraction](https://mlsanity.com/paper/2509.22059/) (2025)
- [Prompting Particle Physics: Tokenized Multi-modal Foundation Models for Combinatorially Many Tasks](https://mlsanity.com/paper/2609.31862/) (2026)
- [Latent Representation Learning in Heavy-Ion Collisions with MaskPoint Transformer](https://mlsanity.com/paper/2510.06691/) (2025)
- [Interaction Concordance Index: Performance Evaluation for Interaction Prediction Methods](https://mlsanity.com/paper/2510.14419/) (2025)
- [Learnable cut flow for high energy physics](https://mlsanity.com/paper/2503.22498/) (2025)
- [Space filling positionality and the Spiroformer](https://mlsanity.com/paper/2507.08456/) (2025)

## Abstract (arXiv)

Physical theories grounded in mathematical symmetries are an essential component of our understanding of a wide range of properties of the universe. Similarly, in the domain of machine learning, an awareness of symmetries such as rotation or permutation invariance has driven impressive performance breakthroughs in computer vision, natural language processing, and other important applications. In this report, we argue that both the physics community and the broader machine learning community have much to understand and potentially to gain from a deeper investment in research concerning symmetry group equivariant machine learning architectures. For some applications, the introduction of symmetries into the fundamental structural design can yield models that are more economical (i.e. contain fewer, but more expressive, learned parameters), interpretable (i.e. more explainable or directly mappable to physical quantities), and/or trainable (i.e. more efficient in both data and computational requirements). We discuss various figures of merit for evaluating these models as well as some potential benefits and limitations of these methods for a variety of physics applications. Research and investment into these approaches will lay the foundation for future architectures that are potentially more robust under new computational paradigms and will provide a richer description of the physical systems to which they are applied.