# Symmetry-Informed Geometric Representation for Molecules, Proteins, and Crystalline Materials

arXiv: https://arxiv.org/abs/2306.09375 · HTML: https://mlsanity.com/paper/2306.09375/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 80 of its 142 references that are in the map.
- First posted on arXiv 15 Jun 2023.

## Builds on (most cited first, 30 of 80)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (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)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Convolutional Networks on Graphs for Learning Molecular Fingerprints](https://mlsanity.com/paper/1509.09292/) (2015)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Recipe for a General, Powerful, Scalable Graph Transformer](https://mlsanity.com/paper/2205.12454/) (2022)
- [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)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [Principal Neighbourhood Aggregation for Graph Nets](https://mlsanity.com/paper/2004.05718/) (2020)
- [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)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [GuacaMol: Benchmarking Models for De Novo Molecular Design](https://mlsanity.com/paper/1811.09621/) (2018)
- [Crystal Diffusion Variational Autoencoder for Periodic Material Generation](https://mlsanity.com/paper/2110.06197/) (2021)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [Torsional Diffusion for Molecular Conformer Generation](https://mlsanity.com/paper/2206.01729/) (2022)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Molecules](https://mlsanity.com/paper/2011.14115/) (2020)

## Built on by (most cited first, 22 of 22)

- [A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systems](https://mlsanity.com/paper/2312.07511/) (2023)
- [Scalable Generation of Spatial Transcriptomics from Histology Images via Whole-Slide Flow Matching](https://mlsanity.com/paper/2506.05361/) (2025)
- [Evaluating representation learning on the protein structure universe](https://mlsanity.com/paper/2406.13864/) (2024)
- [Molecule Joint Auto-Encoding: Trajectory Pretraining with 2D and 3D Diffusion](https://mlsanity.com/paper/2312.03475/) (2023)
- [A Multi-Grained Symmetric Differential Equation Model for Learning Protein-Ligand Binding Dynamics](https://mlsanity.com/paper/2401.15122/) (2024)
- [OBELiX: A Curated Dataset of Crystal Structures and Experimentally Measured Ionic Conductivities for Lithium Solid-State Electrolytes](https://mlsanity.com/paper/2502.14234/) (2025)
- [OrbitAll: A Unified Quantum Mechanical Representation Deep Learning Framework for All Molecular Systems](https://mlsanity.com/paper/2507.03853/) (2025)
- [Building-Block Aware Generative Modeling for 3D Crystals of Metal Organic Frameworks](https://mlsanity.com/paper/2505.08531/) (2025)
- [Flow Along the K-Amplitude for Generative Modeling](https://mlsanity.com/paper/2504.19353/) (2025)
- [Learning Lagrangian Interaction Dynamics with Sampling-Based Model Order Reduction](https://mlsanity.com/paper/2407.03925/) (2024)
- [EDBench: Large-Scale Electron Density Data for Molecular Modeling](https://mlsanity.com/paper/2505.09262/) (2025)
- [MetamatBench: Integrating Heterogeneous Data, Computational Tools, and Visual Interface for Metamaterial Discovery](https://mlsanity.com/paper/2505.20299/) (2025)
- [UniMate: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation](https://mlsanity.com/paper/2506.15722/) (2025)
- [Using GNN property predictors as molecule generators](https://mlsanity.com/paper/2406.03278/) (2024)
- [Manifold-Constrained Nucleus-Level Denoising Diffusion Model for Structure-Based Drug Design](https://mlsanity.com/paper/2409.10584/) (2024)
- [Expert-Guided LLM Reasoning for Battery Discovery: From AI-Driven Hypothesis to Synthesis and Characterization](https://mlsanity.com/paper/2507.16110/) (2025)
- [A Single Architecture for Representing Invariance Under Any Space Group](https://mlsanity.com/paper/2512.13989/) (2025)
- [How Far Can You Grow? Characterizing the Extrapolation Frontier of Graph Generative Models for Materials Science](https://mlsanity.com/paper/2602.09309/) (2026)
- [A quatum inspired neural network for geometric modeling](https://mlsanity.com/paper/2401.01801/) (2024)
- [AlphaFold Database Debiasing for Robust Inverse Folding](https://mlsanity.com/paper/2506.08365/) (2025)
- [Demystify Protein Generation with Hierarchical Conditional Diffusion Models](https://mlsanity.com/paper/2507.18603/) (2025)
- [Rigidity-Aware Geometric Pretraining for Protein Design and Conformational Ensembles](https://mlsanity.com/paper/2603.02406/) (2026)

## Abstract (arXiv)

Artificial intelligence for scientific discovery has recently generated significant interest within the machine learning and scientific communities, particularly in the domains of chemistry, biology, and material discovery. For these scientific problems, molecules serve as the fundamental building blocks, and machine learning has emerged as a highly effective and powerful tool for modeling their geometric structures. Nevertheless, due to the rapidly evolving process of the field and the knowledge gap between science (e.g., physics, chemistry, & biology) and machine learning communities, a benchmarking study on geometrical representation for such data has not been conducted. To address such an issue, in this paper, we first provide a unified view of the current symmetry-informed geometric methods, classifying them into three main categories: invariance, equivariance with spherical frame basis, and equivariance with vector frame basis. Then we propose a platform, coined Geom3D, which enables benchmarking the effectiveness of geometric strategies. Geom3D contains 16 advanced symmetry-informed geometric representation models and 14 geometric pretraining methods over 46 diverse datasets, including small molecules, proteins, and crystalline materials. We hope that Geom3D can, on the one hand, eliminate barriers for machine learning researchers interested in exploring scientific problems; and, on the other hand, provide valuable guidance for researchers in computational chemistry, structural biology, and materials science, aiding in the informed selection of representation techniques for specific applications.