# Space Group Constrained Crystal Generation

arXiv: https://arxiv.org/abs/2402.03992 · HTML: https://mlsanity.com/paper/2402.03992/

## Key facts

- Cited by 73 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 43 references that are in the map.
- First posted on arXiv 6 Feb 2024.

## Builds on (most cited first, 17 of 17)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Argmax Flows and Multinomial Diffusion: Learning Categorical Distributions](https://mlsanity.com/paper/2102.05379/) (2021)
- [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)
- [Symmetry-adapted generation of 3d point sets for the targeted discovery of molecules](https://mlsanity.com/paper/1906.00957/) (2019)
- [MatterGen: a generative model for inorganic materials design](https://mlsanity.com/paper/2312.03687/) (2023)
- [Inverse design of 3d molecular structures with conditional generative neural networks](https://mlsanity.com/paper/2109.04824/) (2021)
- [An invertible crystallographic representation for general inverse design of inorganic crystals with targeted properties](https://mlsanity.com/paper/2005.07609/) (2020)
- [Data-Driven Approach to Encoding and Decoding 3-D Crystal Structures](https://mlsanity.com/paper/1909.00949/) (2019)
- [CrystalGAN: Learning to Discover Crystallographic Structures with Generative Adversarial Networks](https://mlsanity.com/paper/1810.11203/) (2018)
- [Physics Guided Deep Learning for Generative Design of Crystal Materials with Symmetry Constraints](https://mlsanity.com/paper/2203.14352/) (2022)
- [Crystal structure prediction with machine learning-based element substitution](https://mlsanity.com/paper/2201.11188/) (2022)
- [Crystal structure prediction of materials with high symmetry using differential evolution](https://mlsanity.com/paper/2104.09764/) (2021)
- [Vector Field Oriented Diffusion Model for Crystal Material Generation](https://mlsanity.com/paper/2401.05402/) (2023)

## Built on by (most cited first, 30 of 73)

- [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)
- [A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applications](https://mlsanity.com/paper/2403.00485/) (2024)
- [Open Materials Generation with Stochastic Interpolants](https://mlsanity.com/paper/2502.02582/) (2025)
- [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)
- [MOFFlow: Flow Matching for Structure Prediction of Metal-Organic Frameworks](https://mlsanity.com/paper/2410.17270/) (2024)
- [A Periodic Bayesian Flow for Material Generation](https://mlsanity.com/paper/2502.02016/) (2025)
- [Periodic Materials Generation using Text-Guided Joint Diffusion Model](https://mlsanity.com/paper/2503.00522/) (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)
- [Ab Initio Structure Solutions from Nanocrystalline Powder Diffraction Data](https://mlsanity.com/paper/2406.10796/) (2024)
- [All that structure matches does not glitter](https://mlsanity.com/paper/2509.12178/) (2025)
- [AI-driven inverse design of materials: Past, present and future](https://mlsanity.com/paper/2411.09429/) (2024)
- [Materials Generation in the Era of Artificial Intelligence: A Comprehensive Survey](https://mlsanity.com/paper/2505.16379/) (2025)
- [Flexible MOF Generation with Torsion-Aware Flow Matching](https://mlsanity.com/paper/2505.17914/) (2025)
- [Crystalite: A Lightweight Transformer for Efficient Crystal Modeling](https://mlsanity.com/paper/2604.02270/) (2026)
- [Symmetry-Aware Bayesian Flow Networks for Crystal Generation](https://mlsanity.com/paper/2502.03146/) (2025)
- [Kinetic Langevin Diffusion for Crystalline Materials Generation](https://mlsanity.com/paper/2507.03602/) (2025)
- [Siamese Foundation Models for Crystal Structure Prediction](https://mlsanity.com/paper/2503.10471/) (2025)
- [A Survey of AI for Materials Science: Foundation Models, LLM Agents, Datasets, and Tools](https://mlsanity.com/paper/2506.20743/) (2025)
- [Multimodal Crystal Flow: Any-to-Any Modality Generation for Unified Crystal Modeling](https://mlsanity.com/paper/2602.20210/) (2026)
- [Boltzmann generators for amorphous particle systems](https://mlsanity.com/paper/2512.16607/) (2025)
- [Harnessing AtomisticSkills for Agentic Atomistic Research](https://mlsanity.com/paper/2605.24002/) (2026)
- [CrystalGRW: Generative Modeling of Crystal Structures with Targeted Properties via Geodesic Random Walks](https://mlsanity.com/paper/2501.08998/) (2025)
- [DMFlow: Disordered Materials Generation by Flow Matching](https://mlsanity.com/paper/2602.04734/) (2026)
- [Zatom-1: Towards a Multimodal Foundation Model for 3D Molecules and Materials](https://mlsanity.com/paper/2602.22251/) (2026)
- [MiAD: Mirage Atom Diffusion for De Novo Crystal Generation](https://mlsanity.com/paper/2511.14426/) (2025)

## Abstract (arXiv)

Crystals are the foundation of numerous scientific and industrial applications. While various learning-based approaches have been proposed for crystal generation, existing methods seldom consider the space group constraint which is crucial in describing the geometry of crystals and closely relevant to many desirable properties. However, considering space group constraint is challenging owing to its diverse and nontrivial forms. In this paper, we reduce the space group constraint into an equivalent formulation that is more tractable to be handcrafted into the generation process. In particular, we translate the space group constraint into two parts: the basis constraint of the invariant logarithmic space of the lattice matrix and the Wyckoff position constraint of the fractional coordinates. Upon the derived constraints, we then propose DiffCSP++, a novel diffusion model that has enhanced a previous work DiffCSP by further taking space group constraint into account. Experiments on several popular datasets verify the benefit of the involvement of the space group constraint, and show that our DiffCSP++ achieves promising performance on crystal structure prediction, ab initio crystal generation and controllable generation with customized space groups.