# SymmCD: Symmetry-Preserving Crystal Generation with Diffusion Models

arXiv: https://arxiv.org/abs/2502.03638 · HTML: https://mlsanity.com/paper/2502.03638/

## Key facts

- Cited by 51 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 65 references that are in the map.
- First posted on arXiv 5 Feb 2025.

## Builds on (most cited first, 30 of 35)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Crystal Diffusion Variational Autoencoder for Periodic Material Generation](https://mlsanity.com/paper/2110.06197/) (2021)
- [Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Molecules](https://mlsanity.com/paper/2011.14115/) (2020)
- [The Open Catalyst 2020 (OC20) Dataset and Community Challenges](https://mlsanity.com/paper/2010.09990/) (2020)
- [E(n) Equivariant Normalizing Flows](https://mlsanity.com/paper/2105.09016/) (2021)
- [Crystal Structure Prediction by Joint Equivariant Diffusion](https://mlsanity.com/paper/2309.04475/) (2023)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [FlowMM: Generating Materials with Riemannian Flow Matching](https://mlsanity.com/paper/2406.04713/) (2024)
- [MatterGen: a generative model for inorganic materials design](https://mlsanity.com/paper/2312.03687/) (2023)
- [Fine-Tuned Language Models Generate Stable Inorganic Materials as Text](https://mlsanity.com/paper/2402.04379/) (2024)
- [Periodic Graph Transformers for Crystal Material Property Prediction](https://mlsanity.com/paper/2209.11807/) (2022)
- [Space Group Constrained Crystal Generation](https://mlsanity.com/paper/2402.03992/) (2024)
- [A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systems](https://mlsanity.com/paper/2312.07511/) (2023)
- [Language models can generate molecules, materials, and protein binding sites directly in three dimensions as XYZ, CIF, and PDB files](https://mlsanity.com/paper/2305.05708/) (2023)
- [Scalable Diffusion for Materials Generation](https://mlsanity.com/paper/2311.09235/) (2023)
- [Space Group Informed Transformer for Crystalline Materials Generation](https://mlsanity.com/paper/2403.15734/) (2024)
- [Matbench Discovery -- A framework to evaluate machine learning crystal stability predictions](https://mlsanity.com/paper/2308.14920/) (2023)
- [CrystalGAN: Learning to Discover Crystallographic Structures with Generative Adversarial Networks](https://mlsanity.com/paper/1810.11203/) (2018)
- [Crystal-GFN: sampling crystals with desirable properties and constraints](https://mlsanity.com/paper/2310.04925/) (2023)
- [Towards Symmetry-Aware Generation of Periodic Materials](https://mlsanity.com/paper/2307.02707/) (2023)
- [Symmetry Breaking and Equivariant Neural Networks](https://mlsanity.com/paper/2312.09016/) (2023)
- [Equivariant Networks for Crystal Structures](https://mlsanity.com/paper/2211.15420/) (2022)

## Built on by (most cited first, 30 of 51)

- [Wyckoff Transformer: Generation of Symmetric Crystals](https://mlsanity.com/paper/2503.02407/) (2025)
- [WyckoffDiff -- A Generative Diffusion Model for Crystal Symmetry](https://mlsanity.com/paper/2502.06485/) (2025)
- [LeMat-GenBench: A Unified Evaluation Framework for Crystal Generative Models](https://mlsanity.com/paper/2512.04562/) (2025)
- [Space Group Equivariant Crystal Diffusion](https://mlsanity.com/paper/2505.10994/) (2025)
- [PLaID++: A Preference Aligned Language Model for Targeted Inorganic Materials Design](https://mlsanity.com/paper/2509.07150/) (2025)
- [LLM Meets Diffusion: A Hybrid Framework for Crystal Material Generation](https://mlsanity.com/paper/2510.23040/) (2025)
- [Materials Generation in the Era of Artificial Intelligence: A Comprehensive Survey](https://mlsanity.com/paper/2505.16379/) (2025)
- [Crystalite: A Lightweight Transformer for Efficient Crystal Modeling](https://mlsanity.com/paper/2604.02270/) (2026)
- [Kinetic Langevin Diffusion for Crystalline Materials Generation](https://mlsanity.com/paper/2507.03602/) (2025)
- [Artificial Intelligence and Generative Models for Materials Discovery -- A Review](https://mlsanity.com/paper/2508.03278/) (2025)
- [Multimodal Crystal Flow: Any-to-Any Modality Generation for Unified Crystal Modeling](https://mlsanity.com/paper/2602.20210/) (2026)
- [CrystalDiT: A Diffusion Transformer for Crystal Generation](https://mlsanity.com/paper/2508.16614/) (2025)
- [Continuous SUN (Stable, Unique, and Novel) Metric for Generative Modeling of Inorganic Crystals](https://mlsanity.com/paper/2510.12405/) (2025)
- [Boltzmann generators for amorphous particle systems](https://mlsanity.com/paper/2512.16607/) (2025)
- [DMFlow: Disordered Materials Generation by Flow Matching](https://mlsanity.com/paper/2602.04734/) (2026)
- [MolCrystalFlow: Molecular Crystal Structure Prediction via Flow Matching](https://mlsanity.com/paper/2602.16020/) (2026)
- [Zatom-1: Towards a Multimodal Foundation Model for 3D Molecules and Materials](https://mlsanity.com/paper/2602.22251/) (2026)
- [CrystalGym: A New Benchmark for Materials Discovery Using Reinforcement Learning](https://mlsanity.com/paper/2509.23156/) (2025)
- [MiAD: Mirage Atom Diffusion for De Novo Crystal Generation](https://mlsanity.com/paper/2511.14426/) (2025)
- [Generative Models for Crystalline Materials](https://mlsanity.com/paper/2511.22652/) (2025)
- [OXtal: An All-Atom Diffusion Model for Organic Crystal Structure Prediction](https://mlsanity.com/paper/2512.06987/) (2025)
- [Crys-JEPA: Accelerating Crystal Discovery via Embedding Screening and Generative Refinement](https://mlsanity.com/paper/2605.14759/) (2026)
- [CLOUD: A Scalable and Physics-Informed Foundation Model for Crystal Representation Learning](https://mlsanity.com/paper/2506.17345/) (2025)
- [Fast Organic Crystal Structure Prediction with Unit Cell Flow Matching](https://mlsanity.com/paper/2606.03199/) (2026)
- [AI-Assisted Rapid Crystal Structure Generation Towards a Target Local Environment](https://mlsanity.com/paper/2506.08224/) (2025)
- [SLayerGen: a Crystal Generative Model for all Space and Layer Groups](https://mlsanity.com/paper/2605.08262/) (2026)
- [CrystalREPA: Transferring Physical Priors from Universal MLIPs to Crystal Generative Models](https://mlsanity.com/paper/2605.08960/) (2026)
- [How Far Can You Grow? Characterizing the Extrapolation Frontier of Graph Generative Models for Materials Science](https://mlsanity.com/paper/2602.09309/) (2026)
- [Fourier Transformers for Latent Crystallographic Diffusion and Generative Modeling](https://mlsanity.com/paper/2602.12045/) (2026)
- [Composable Crystals: Controllable Materials Discovery via Concept Learning](https://mlsanity.com/paper/2605.14769/) (2026)

## Abstract (arXiv)

Generating novel crystalline materials has the potential to lead to advancements in fields such as electronics, energy storage, and catalysis. The defining characteristic of crystals is their symmetry, which plays a central role in determining their physical properties. However, existing crystal generation methods either fail to generate materials that display the symmetries of real-world crystals, or simply replicate the symmetry information from examples in a database. To address this limitation, we propose SymmCD, a novel diffusion-based generative model that explicitly incorporates crystallographic symmetry into the generative process. We decompose crystals into two components and learn their joint distribution through diffusion: 1) the asymmetric unit, the smallest subset of the crystal which can generate the whole crystal through symmetry transformations, and; 2) the symmetry transformations needed to be applied to each atom in the asymmetric unit. We also use a novel and interpretable representation for these transformations, enabling generalization across different crystallographic symmetry groups. We showcase the competitive performance of SymmCD on a subset of the Materials Project, obtaining diverse and valid crystals with realistic symmetries and predicted properties.