# A Periodic Bayesian Flow for Material Generation

arXiv: https://arxiv.org/abs/2502.02016 · HTML: https://mlsanity.com/paper/2502.02016/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 60 of its 121 references that are in the map.
- First posted on arXiv 4 Feb 2025.

## Builds on (most cited first, 30 of 60)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [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)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Variational Graph Auto-Encoders](https://mlsanity.com/paper/1611.07308/) (2016)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [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)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking](https://mlsanity.com/paper/2210.01776/) (2022)
- [Crystal Diffusion Variational Autoencoder for Periodic Material Generation](https://mlsanity.com/paper/2110.06197/) (2021)
- [Torsional Diffusion for Molecular Conformer Generation](https://mlsanity.com/paper/2206.01729/) (2022)
- [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)
- [Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problem](https://mlsanity.com/paper/2206.04119/) (2022)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Geometric Latent Diffusion Models for 3D Molecule Generation](https://mlsanity.com/paper/2305.01140/) (2023)

## Built on by (most cited first, 16 of 16)

- [Space Group Equivariant Crystal Diffusion](https://mlsanity.com/paper/2505.10994/) (2025)
- [LLM Meets Diffusion: A Hybrid Framework for Crystal Material Generation](https://mlsanity.com/paper/2510.23040/) (2025)
- [All that structure matches does not glitter](https://mlsanity.com/paper/2509.12178/) (2025)
- [Materials Generation in the Era of Artificial Intelligence: A Comprehensive Survey](https://mlsanity.com/paper/2505.16379/) (2025)
- [Open Materials Generation with Inference-Time Reinforcement Learning](https://mlsanity.com/paper/2602.00424/) (2026)
- [Multimodal Crystal Flow: Any-to-Any Modality Generation for Unified Crystal Modeling](https://mlsanity.com/paper/2602.20210/) (2026)
- [DMFlow: Disordered Materials Generation by Flow Matching](https://mlsanity.com/paper/2602.04734/) (2026)
- [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)
- [SLayerGen: a Crystal Generative Model for all Space and Layer Groups](https://mlsanity.com/paper/2605.08262/) (2026)
- [MolPIF: A Parameter Interpolation Flow Model for Molecule Generation](https://mlsanity.com/paper/2507.13762/) (2025)
- [Unsupervised anomaly detection using Bayesian flow networks: application to brain FDG PET in the context of Alzheimer's disease](https://mlsanity.com/paper/2507.17486/) (2025)
- [CrystalGRPO: Target-Aligned and Coverage-Preserving Reinforcement Learning for Flow-Based Crystal Structure Prediction](https://mlsanity.com/paper/2608.06582/) (2026)
- [Packora: Systematic Design for Generative Molecular Crystal Structure Prediction](https://mlsanity.com/paper/2608.26962/) (2026)
- [EP-Flow: Disordered Crystal Structure Prediction without Site-Level Annotations](https://mlsanity.com/paper/2610.01315/) (2026)
- [AnchorPose for Geometry-Aware MOF Assembly through Meso-Grained Pose Generation](https://mlsanity.com/paper/2610.05843/) (2026)

## Abstract (arXiv)

Generative modeling of crystal data distribution is an important yet challenging task due to the unique periodic physical symmetry of crystals. Diffusion-based methods have shown early promise in modeling crystal distribution. More recently, Bayesian Flow Networks were introduced to aggregate noisy latent variables, resulting in a variance-reduced parameter space that has been shown to be advantageous for modeling Euclidean data distributions with structural constraints (Song et al., 2023). Inspired by this, we seek to unlock its potential for modeling variables located in non-Euclidean manifolds e.g. those within crystal structures, by overcoming challenging theoretical issues. We introduce CrysBFN, a novel crystal generation method by proposing a periodic Bayesian flow, which essentially differs from the original Gaussian-based BFN by exhibiting non-monotonic entropy dynamics. To successfully realize the concept of periodic Bayesian flow, CrysBFN integrates a new entropy conditioning mechanism and empirically demonstrates its significance compared to time-conditioning. Extensive experiments over both crystal ab initio generation and crystal structure prediction tasks demonstrate the superiority of CrysBFN, which consistently achieves new state-of-the-art on all benchmarks. Surprisingly, we found that CrysBFN enjoys a significant improvement in sampling efficiency, e.g., ~100x speedup 10 v.s. 2000 steps network forwards) compared with previous diffusion-based methods on MP-20 dataset. Code is available at https://github.com/wu-han-lin/CrysBFN.