# WyckoffDiff -- A Generative Diffusion Model for Crystal Symmetry

arXiv: https://arxiv.org/abs/2502.06485 · HTML: https://mlsanity.com/paper/2502.06485/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 18 of its 47 references that are in the map.
- First posted on arXiv 10 Feb 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 18 of 18)

- [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)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution](https://mlsanity.com/paper/2310.16834/) (2023)
- [Argmax Flows and Multinomial Diffusion: Learning Categorical Distributions](https://mlsanity.com/paper/2102.05379/) (2021)
- [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022)
- [A Continuous Time Framework for Discrete Denoising Models](https://mlsanity.com/paper/2205.14987/) (2022)
- [Crystal Diffusion Variational Autoencoder for Periodic Material Generation](https://mlsanity.com/paper/2110.06197/) (2021)
- [Fr'echet ChemNet Distance: A metric for generative models for molecules in drug discovery](https://mlsanity.com/paper/1803.09518/) (2018)
- [Score-based Continuous-time Discrete Diffusion Models](https://mlsanity.com/paper/2211.16750/) (2022)
- [Crystal Structure Prediction by Joint Equivariant Diffusion](https://mlsanity.com/paper/2309.04475/) (2023)
- [Dirichlet Flow Matching with Applications to DNA Sequence Design](https://mlsanity.com/paper/2402.05841/) (2024)
- [Space Group Constrained Crystal Generation](https://mlsanity.com/paper/2402.03992/) (2024)
- [SymmCD: Symmetry-Preserving Crystal Generation with Diffusion Models](https://mlsanity.com/paper/2502.03638/) (2025)

## Built on by (most cited first, 26 of 26)

- [Space Group Equivariant Crystal Diffusion](https://mlsanity.com/paper/2505.10994/) (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)
- [PowerGrow: Feasible Co-Growth of Structures and Dynamics for Power Grid Synthesis](https://mlsanity.com/paper/2509.12212/) (2025)
- [DMFlow: Disordered Materials Generation by Flow Matching](https://mlsanity.com/paper/2602.04734/) (2026)
- [Generative Models for Crystalline Materials](https://mlsanity.com/paper/2511.22652/) (2025)
- [PolyGraph Discrepancy: a classifier-based metric for graph generation](https://mlsanity.com/paper/2510.06122/) (2025)
- [CrystalReasoner: Reasoning and RL for Property-Conditioned Crystal Structure Generation](https://mlsanity.com/paper/2605.14344/) (2026)
- [Transport Novelty Distance: A Distributional Metric for Evaluating Material Generative Models](https://mlsanity.com/paper/2512.09514/) (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)
- [Toward Controllable Catalyst Inverse Design via Large-Scale Autoregressive Pretraining](https://mlsanity.com/paper/2606.17445/) (2026)
- [Discovering Crystal Structure Prediction Algorithms with an AI Co-Scientist](https://mlsanity.com/paper/2606.22866/) (2026)
- [CrystalGRPO: Target-Aligned and Coverage-Preserving Reinforcement Learning for Flow-Based Crystal Structure Prediction](https://mlsanity.com/paper/2608.06582/) (2026)
- [3D Shape Generation: A Survey](https://mlsanity.com/paper/2506.22678/) (2025)
- [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)
- [LapidaryEngine: Fully Conversational Crystal Generation](https://mlsanity.com/paper/2606.14215/) (2026)
- [Context-weighted Discrete Flow Matching](https://mlsanity.com/paper/2607.21427/) (2026)
- [Representations from Pretrained Machine-Learning Interatomic Potentials as Coarse Coordinates for Material Generation and Evaluation](https://mlsanity.com/paper/2607.28776/) (2026)
- [DynaCrys: Crystal Generation with Dynamic Space-Group Diffusion](https://mlsanity.com/paper/2608.07401/) (2026)
- [uFlowCSP: Crystal Structure Prediction using Mean flow generative models](https://mlsanity.com/paper/2609.09799/) (2026)
- [Topology-Stratified Materials Discovery with A Flow-Based Generative Model](https://mlsanity.com/paper/2609.26547/) (2026)
- [GLASS: Global Latent Aggregation with Slot-based Set Decoding for Scalable All-Atom Crystal Generation](https://mlsanity.com/paper/2609.37158/) (2026)

## Abstract (arXiv)

Crystalline materials often exhibit a high level of symmetry. However, most generative models do not account for symmetry, but rather model each atom without any constraints on its position or element. We propose a generative model, Wyckoff Diffusion (WyckoffDiff), which generates symmetry-based descriptions of crystals. This is enabled by considering a crystal structure representation that encodes all symmetry, and we design a novel neural network architecture which enables using this representation inside a discrete generative model framework. In addition to respecting symmetry by construction, the discrete nature of our model enables fast generation. We additionally present a new metric, Fr'echet Wrenformer Distance, which captures the symmetry aspects of the materials generated, and we benchmark WyckoffDiff against recently proposed generative models for crystal generation. As a proof-of-concept study, we use WyckoffDiff to find new materials below the convex hull of thermodynamical stability.