# Space Group Equivariant Crystal Diffusion

arXiv: https://arxiv.org/abs/2505.10994 · HTML: https://mlsanity.com/paper/2505.10994/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 110 references that are in the map.
- First posted on arXiv 16 May 2025.

## Builds on (most cited first, 30 of 43)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [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)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Flow Network based Generative Models for Non-Iterative Diverse Candidate Generation](https://mlsanity.com/paper/2106.04399/) (2021)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [Crystal Diffusion Variational Autoencoder for Periodic Material Generation](https://mlsanity.com/paper/2110.06197/) (2021)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Central Moment Discrepancy (CMD) for Domain-Invariant Representation Learning](https://mlsanity.com/paper/1702.08811/) (2017)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [Crystal Structure Prediction by Joint Equivariant Diffusion](https://mlsanity.com/paper/2309.04475/) (2023)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [FlowMM: Generating Materials with Riemannian Flow Matching](https://mlsanity.com/paper/2406.04713/) (2024)
- [Fine-Tuned Language Models Generate Stable Inorganic Materials as Text](https://mlsanity.com/paper/2402.04379/) (2024)
- [Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction](https://mlsanity.com/paper/2502.12147/) (2025)
- [FAENet: Frame Averaging Equivariant GNN for Materials Modeling](https://mlsanity.com/paper/2305.05577/) (2023)
- [Space Group Constrained Crystal Generation](https://mlsanity.com/paper/2402.03992/) (2024)
- [Bayesian Flow Networks](https://mlsanity.com/paper/2308.07037/) (2023)
- [Scalable Diffusion for Materials Generation](https://mlsanity.com/paper/2311.09235/) (2023)
- [FlowLLM: Flow Matching for Material Generation with Large Language Models as Base Distributions](https://mlsanity.com/paper/2410.23405/) (2024)
- [SymmCD: Symmetry-Preserving Crystal Generation with Diffusion Models](https://mlsanity.com/paper/2502.03638/) (2025)
- [Equivariant Hamiltonian Flows](https://mlsanity.com/paper/1909.13739/) (2019)

## Built on by (most cited first, 15 of 15)

- [LeMat-GenBench: A Unified Evaluation Framework for Crystal Generative Models](https://mlsanity.com/paper/2512.04562/) (2025)
- [Space Group Conditional Flow Matching](https://mlsanity.com/paper/2509.23822/) (2025)
- [Multimodal Crystal Flow: Any-to-Any Modality Generation for Unified Crystal Modeling](https://mlsanity.com/paper/2602.20210/) (2026)
- [Crys-JEPA: Accelerating Crystal Discovery via Embedding Screening and Generative Refinement](https://mlsanity.com/paper/2605.14759/) (2026)
- [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)
- [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)
- [Composable Crystals: Controllable Materials Discovery via Concept Learning](https://mlsanity.com/paper/2605.14769/) (2026)
- [OMatG-flash: An All-Atom Flow Map with Reinforce Adjoint Matching for Scalable Materials Discovery](https://mlsanity.com/paper/2609.26402/) (2026)
- [OrgFlow: Generative Modeling of Organic Crystal Structures from Molecular Graphs](https://mlsanity.com/paper/2602.20195/) (2026)
- [Generating Symmetric Materials using Latent Flow Matching](https://mlsanity.com/paper/2605.10115/) (2026)
- [PRISMat: Policy-Driven, Permutation-Invariant Autoregressive Material Generation](https://mlsanity.com/paper/2605.16612/) (2026)
- [DynaCrys: Crystal Generation with Dynamic Space-Group Diffusion](https://mlsanity.com/paper/2608.07401/) (2026)
- [Symmetry-Breaking De Novo Crystal Generation via Markovian Jump Diffusion](https://mlsanity.com/paper/2608.13457/) (2026)

## Abstract (arXiv)

Accelerating inverse design of crystalline materials with generative models has significant implications for a range of technologies. Unlike other atomic systems, 3D crystals are invariant to discrete groups of isometries called the space groups. Crucially, these space group symmetries are known to heavily influence materials properties. We propose SGEquiDiff, a crystal generative model which naturally handles space group constraints with space group invariant likelihoods. SGEquiD-iff consists of an SE(3)-invariant, telescoping discrete sampler of crystal lattices; permutation-invariant, transformer-based autoregressive sampling of Wyckoff positions, elements, and numbers of symmetrically unique atoms; and space group equivariant diffusion of atomic coordinates. We show that space group equivariant vector fields automatically live in the tangent spaces of the Wyckoff positions. SGEquiDiff achieves state-of-the-art performance on standard benchmark datasets as assessed by quantitative proxy metrics and quantum mechanical calculations. Our code is available at https://github.com/rees-c/sgequidiff.