# A new perspective on building efficient and expressive 3D equivariant graph neural networks

arXiv: https://arxiv.org/abs/2304.04757 · HTML: https://mlsanity.com/paper/2304.04757/

## Key facts

- Cited by 47 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 64 references that are in the map.
- First posted on arXiv 7 Apr 2023.

## Builds on (most cited first, 30 of 48)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [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)
- [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)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (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)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [Analyzing Learned Molecular Representations for Property Prediction](https://mlsanity.com/paper/1904.01561/) (2019)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [Learning from Protein Structure with Geometric Vector Perceptrons](https://mlsanity.com/paper/2009.01411/) (2020)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Structure-based Drug Design with Equivariant Diffusion Models](https://mlsanity.com/paper/2210.13695/) (2022)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Neural Descriptor Fields: SE(3)-Equivariant Object Representations for Manipulation](https://mlsanity.com/paper/2112.05124/) (2021)
- [Spherical Message Passing for 3D Graph Networks](https://mlsanity.com/paper/2102.05013/) (2021)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)

## Built on by (most cited first, 30 of 47)

- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Transferable Boltzmann Generators](https://mlsanity.com/paper/2406.14426/) (2024)
- [Equivariant Frames and the Impossibility of Continuous Canonicalization](https://mlsanity.com/paper/2402.16077/) (2024)
- [A Group Symmetric Stochastic Differential Equation Model for Molecule Multi-modal Pretraining](https://mlsanity.com/paper/2305.18407/) (2023)
- [Accurate transition state generation with an object-aware equivariant elementary reaction diffusion model](https://mlsanity.com/paper/2304.06174/) (2023)
- [A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applications](https://mlsanity.com/paper/2403.00485/) (2024)
- [Uncovering Neural Scaling Laws in Molecular Representation Learning](https://mlsanity.com/paper/2309.15123/) (2023)
- [Symmetry-Informed Geometric Representation for Molecules, Proteins, and Crystalline Materials](https://mlsanity.com/paper/2306.09375/) (2023)
- [Generalist Equivariant Transformer Towards 3D Molecular Interaction Learning](https://mlsanity.com/paper/2306.01474/) (2023)
- [Beyond Canonicalization: How Tensorial Messages Improve Equivariant Message Passing](https://mlsanity.com/paper/2405.15389/) (2024)
- [Lorentz Local Canonicalization: How to Make Any Network Lorentz-Equivariant](https://mlsanity.com/paper/2505.20280/) (2025)
- [On the Completeness of Invariant Geometric Deep Learning Models](https://mlsanity.com/paper/2402.04836/) (2024)
- [Molecule Joint Auto-Encoding: Trajectory Pretraining with 2D and 3D Diffusion](https://mlsanity.com/paper/2312.03475/) (2023)
- [AI-driven inverse design of materials: Past, present and future](https://mlsanity.com/paper/2411.09429/) (2024)
- [Learning Over Molecular Conformer Ensembles: Datasets and Benchmarks](https://mlsanity.com/paper/2310.00115/) (2023)
- [MolSpectra: Pre-training 3D Molecular Representation with Multi-modal Energy Spectra](https://mlsanity.com/paper/2502.16284/) (2025)
- [On the Expressive Power of Sparse Geometric MPNNs](https://mlsanity.com/paper/2407.02025/) (2024)
- [MatLLMSearch: Crystal Structure Discovery with Evolution-Guided Large Language Models](https://mlsanity.com/paper/2502.20933/) (2025)
- [MatRIS: Toward Reliable and Efficient Pretrained Machine Learning Interatomic Potentials](https://mlsanity.com/paper/2603.02002/) (2026)
- [Equivariant Graph Network Approximations of High-Degree Polynomials for Force Field Prediction](https://mlsanity.com/paper/2411.04219/) (2024)
- [DuSEGO: Dual Second-order Equivariant Graph Ordinary Differential Equation](https://mlsanity.com/paper/2411.10000/) (2024)
- [Structure-Aware E(3)-Invariant Molecular Conformer Aggregation Networks](https://mlsanity.com/paper/2402.01975/) (2024)
- [Building-Block Aware Generative Modeling for 3D Crystals of Metal Organic Frameworks](https://mlsanity.com/paper/2505.08531/) (2025)
- [Geometric Hyena Networks for Large-scale Equivariant Learning](https://mlsanity.com/paper/2505.22560/) (2025)
- [React-OT: Optimal Transport for Generating Transition State in Chemical Reactions](https://mlsanity.com/paper/2404.13430/) (2024)
- [Group Downsampling with Equivariant Anti-aliasing](https://mlsanity.com/paper/2504.17258/) (2025)
- [Tokenizing Electron Cloud in Protein-Ligand Interaction Learning](https://mlsanity.com/paper/2505.19014/) (2025)
- [SE3Set: Harnessing equivariant hypergraph neural networks for molecular representation learning](https://mlsanity.com/paper/2405.16511/) (2024)
- [AlphaNet: Scaling Up Local-frame-based Atomistic Interatomic Potential](https://mlsanity.com/paper/2501.07155/) (2025)
- [Topotein: Topological Deep Learning for Protein Representation Learning](https://mlsanity.com/paper/2509.03885/) (2025)

## Abstract (arXiv)

Geometric deep learning enables the encoding of physical symmetries in modeling 3D objects. Despite rapid progress in encoding 3D symmetries into Graph Neural Networks (GNNs), a comprehensive evaluation of the expressiveness of these networks through a local-to-global analysis lacks today. In this paper, we propose a local hierarchy of 3D isomorphism to evaluate the expressive power of equivariant GNNs and investigate the process of representing global geometric information from local patches. Our work leads to two crucial modules for designing expressive and efficient geometric GNNs; namely local substructure encoding (LSE) and frame transition encoding (FTE). To demonstrate the applicability of our theory, we propose LEFTNet which effectively implements these modules and achieves state-of-the-art performance on both scalar-valued and vector-valued molecular property prediction tasks. We further point out the design space for future developments of equivariant graph neural networks. Our codes are available at urlhttps://github.com/yuanqidu/LeftNet.