# Automatic Symmetry Discovery with Lie Algebra Convolutional Network

arXiv: https://arxiv.org/abs/2109.07103 · HTML: https://mlsanity.com/paper/2109.07103/

## Key facts

- Cited by 83 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 49 references that are in the map.
- First posted on arXiv 15 Sep 2021.

## Builds on (most cited first, 28 of 28)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [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)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [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)
- [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)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [On Learning Sets of Symmetric Elements](https://mlsanity.com/paper/2002.08599/) (2020)
- [Neural Mechanics: Symmetry and Broken Conservation Laws in Deep Learning Dynamics](https://mlsanity.com/paper/2012.04728/) (2020)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [Meta-Learning Symmetries by Reparameterization](https://mlsanity.com/paper/2007.02933/) (2020)
- [Deep Models of Interactions Across Sets](https://mlsanity.com/paper/1803.02879/) (2018)
- [Learning Invariances in Neural Networks](https://mlsanity.com/paper/2010.11882/) (2020)
- [Universal Equivariant Multilayer Perceptrons](https://mlsanity.com/paper/2002.02912/) (2020)
- [Learning the Irreducible Representations of Commutative Lie Groups](https://mlsanity.com/paper/1402.4437/) (2014)
- [Disentangling by Subspace Diffusion](https://mlsanity.com/paper/2006.12982/) (2020)

## Built on by (most cited first, 30 of 83)

- [On scientific understanding with artificial intelligence](https://mlsanity.com/paper/2204.01467/) (2022)
- [Learning Symmetric Embeddings for Equivariant World Models](https://mlsanity.com/paper/2204.11371/) (2022)
- [Learning Partial Equivariances from Data](https://mlsanity.com/paper/2110.10211/) (2021)
- [Generative Adversarial Symmetry Discovery](https://mlsanity.com/paper/2302.00236/) (2023)
- [The Surprising Effectiveness of Equivariant Models in Domains with Latent Symmetry](https://mlsanity.com/paper/2211.09231/) (2022)
- [LieGG: Studying Learned Lie Group Generators](https://mlsanity.com/paper/2210.04345/) (2022)
- [Latent Space Symmetry Discovery](https://mlsanity.com/paper/2310.00105/) (2023)
- [Equivariant Networks for Crystal Structures](https://mlsanity.com/paper/2211.15420/) (2022)
- [Learning Layer-wise Equivariances Automatically using Gradients](https://mlsanity.com/paper/2310.06131/) (2023)
- [Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolution](https://mlsanity.com/paper/2310.02299/) (2023)
- [A General Theory of Correct, Incorrect, and Extrinsic Equivariance](https://mlsanity.com/paper/2303.04745/) (2023)
- [Learning to Extrapolate: A Transductive Approach](https://mlsanity.com/paper/2304.14329/) (2023)
- [Homomorphism Autoencoder -- Learning Group Structured Representations from Observed Transitions](https://mlsanity.com/paper/2207.12067/) (2022)
- [Structure-preserving GANs](https://mlsanity.com/paper/2202.01129/) (2022)
- [Learning Infinitesimal Generators of Continuous Symmetries from Data](https://mlsanity.com/paper/2410.21853/) (2024)
- [Regularizing Towards Soft Equivariance Under Mixed Symmetries](https://mlsanity.com/paper/2306.00356/) (2023)
- [Unsupervised Learning of Equivariant Structure from Sequences](https://mlsanity.com/paper/2210.05972/) (2022)
- [Symmetry-Informed Governing Equation Discovery](https://mlsanity.com/paper/2405.16756/) (2024)
- [Symmetry Discovery Beyond Affine Transformations](https://mlsanity.com/paper/2406.03619/) (2024)
- [On genuine invariance learning without weight-tying](https://mlsanity.com/paper/2308.03904/) (2023)
- [FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantities](https://mlsanity.com/paper/2210.00272/) (2022)
- [Exploiting Code Symmetries for Learning Program Semantics](https://mlsanity.com/paper/2308.03312/) (2023)
- [Almost Equivariance via Lie Algebra Convolutions](https://mlsanity.com/paper/2310.13164/) (2023)
- [A Generative Model of Symmetry Transformations](https://mlsanity.com/paper/2403.01946/) (2024)
- [Policy Gradient Methods in the Presence of Symmetries and State Abstractions](https://mlsanity.com/paper/2305.05666/) (2023)
- [Symmetry Discovery for Different Data Types](https://mlsanity.com/paper/2410.09841/) (2024)
- [Diffusion Generative Modeling on Lie Group Representations](https://mlsanity.com/paper/2502.02513/) (2025)
- [A Unified Framework for Discovering Discrete Symmetries](https://mlsanity.com/paper/2309.02898/) (2023)
- [Achieving Approximate Symmetry Is Exponentially Easier than Exact Symmetry](https://mlsanity.com/paper/2512.11855/) (2025)
- [Hamiltonian Neural Networks with Automatic Symmetry Detection](https://mlsanity.com/paper/2301.07928/) (2023)

## Abstract (arXiv)

Existing equivariant neural networks require prior knowledge of the symmetry group and discretization for continuous groups. We propose to work with Lie algebras (infinitesimal generators) instead of Lie groups. Our model, the Lie algebra convolutional network (L-conv) can automatically discover symmetries and does not require discretization of the group. We show that L-conv can serve as a building block to construct any group equivariant feedforward architecture. Both CNNs and Graph Convolutional Networks can be expressed as L-conv with appropriate groups. We discover direct connections between L-conv and physics: (1) group invariant loss generalizes field theory (2) Euler-Lagrange equation measures the robustness, and (3) equivariance leads to conservation laws and Noether current.These connections open up new avenues for designing more general equivariant networks and applying them to important problems in physical sciences