# Equivariant Networks for Crystal Structures

arXiv: https://arxiv.org/abs/2211.15420 · HTML: https://mlsanity.com/paper/2211.15420/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 95 references that are in the map.
- First posted on arXiv 15 Nov 2022.

## Builds on (most cited first, 30 of 39)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [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)
- [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)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [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)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (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)
- [Crystal Diffusion Variational Autoencoder for Periodic Material Generation](https://mlsanity.com/paper/2110.06197/) (2021)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [The Open Catalyst 2020 (OC20) Dataset and Community Challenges](https://mlsanity.com/paper/2010.09990/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Generalization and Representational Limits of Graph Neural Networks](https://mlsanity.com/paper/2002.06157/) (2020)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [E(n) Equivariant Normalizing Flows](https://mlsanity.com/paper/2105.09016/) (2021)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Machine learning for molecular simulation](https://mlsanity.com/paper/1911.02792/) (2019)
- [On Learning Sets of Symmetric Elements](https://mlsanity.com/paper/2002.08599/) (2020)

## Built on by (most cited first, 24 of 24)

- [A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systems](https://mlsanity.com/paper/2312.07511/) (2023)
- [SymmCD: Symmetry-Preserving Crystal Generation with Diffusion Models](https://mlsanity.com/paper/2502.03638/) (2025)
- [LLM-Prop: Predicting Physical And Electronic Properties Of Crystalline Solids From Their Text Descriptions](https://mlsanity.com/paper/2310.14029/) (2023)
- [A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applications](https://mlsanity.com/paper/2403.00485/) (2024)
- [Symmetry Breaking and Equivariant Neural Networks](https://mlsanity.com/paper/2312.09016/) (2023)
- [Improving Equivariant Networks with Probabilistic Symmetry Breaking](https://mlsanity.com/paper/2503.21985/) (2025)
- [MatSciML: A Broad, Multi-Task Benchmark for Solid-State Materials Modeling](https://mlsanity.com/paper/2309.05934/) (2023)
- [AI-driven inverse design of materials: Past, present and future](https://mlsanity.com/paper/2411.09429/) (2024)
- [Implicit Convolutional Kernels for Steerable CNNs](https://mlsanity.com/paper/2212.06096/) (2022)
- [M2Hub: Unlocking the Potential of Machine Learning for Materials Discovery](https://mlsanity.com/paper/2307.05378/) (2023)
- [Improved Canonicalization for Model Agnostic Equivariance](https://mlsanity.com/paper/2405.14089/) (2024)
- [Supra-Laplacian Encoding for Transformer on Dynamic Graphs](https://mlsanity.com/paper/2409.17986/) (2024)
- [Gauge-Equivariant Graph Neural Networks for Lattice Gauge Theories](https://mlsanity.com/paper/2604.20797/) (2026)
- [Densely Connected G-invariant Deep Neural Networks with Signed Permutation Representations](https://mlsanity.com/paper/2303.04614/) (2023)
- [Equivariant Neural Networks for Force-Field Models of Lattice Systems](https://mlsanity.com/paper/2601.04104/) (2026)
- [Equivariant Parameter Sharing for Porous Crystalline Materials](https://mlsanity.com/paper/2304.01628/) (2023)
- [Graph neural network force fields for adiabatic dynamics of lattice Hamiltonians](https://mlsanity.com/paper/2603.02039/) (2026)
- [Graph Neural Networks for Carbon Dioxide Adsorption Prediction in Aluminium-Exchanged Zeolites](https://mlsanity.com/paper/2403.12659/) (2024)
- [Symmetry-Informed Graph Neural Networks for Carbon Dioxide Isotherm and Adsorption Prediction in Aluminum-Substituted Zeolites](https://mlsanity.com/paper/2503.22737/) (2025)
- [Quotient Complex Transformer (QCformer) for Perovskite Data Analysis](https://mlsanity.com/paper/2505.09174/) (2025)
- [Fundamental Limits of Crystalline Equivariant Graph Neural Networks: A Circuit Complexity Perspective](https://mlsanity.com/paper/2510.05494/) (2025)
- [High-Throughput Unsupervised Profiling of the Morphology of 316L Powder Particles for Use in Additive Manufacturing](https://mlsanity.com/paper/2512.06012/) (2025)
- [Magnetic HIP-NN for spin dynamics in disordered itinerant magnets](https://mlsanity.com/paper/2606.10349/) (2026)
- [Graph Neural Network Force Fields for Spin Dynamics in Metallic Magnets](https://mlsanity.com/paper/2607.28537/) (2026)

## Abstract (arXiv)

Supervised learning with deep models has tremendous potential for applications in materials science. Recently, graph neural networks have been used in this context, drawing direct inspiration from models for molecules. However, materials are typically much more structured than molecules, which is a feature that these models do not leverage. In this work, we introduce a class of models that are equivariant with respect to crystalline symmetry groups. We do this by defining a generalization of the message passing operations that can be used with more general permutation groups, or that can alternatively be seen as defining an expressive convolution operation on the crystal graph. Empirically, these models achieve competitive results with state-of-the-art on property prediction tasks.