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A Periodic Bayesian Flow for Material Generation
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Introduces CrysBFN, a Bayesian flow network over the hyper-torus with von Mises distributions and entropy conditioning, for periodic E(3)-equivariant crystal generation and structure prediction, achieving SOTA with ~100x fewer sampling steps than diffusion.
Problem
Generative modeling of crystal structures is hard because fractional coordinates live on a periodic (hypertorus) manifold, where Gaussian-based Bayesian flow networks and wrapped-normal diffusion introduce bias and inefficiency.
Approach
Derives a periodic Bayesian flow on the hypertorus using von Mises distributions, which breaks additive accuracy, so the receiver's concentration is conditioned on as an entropy signal instead of time; adds a closed-form non-iterative sampler for the flow distribution and a numerically searched linear-entropy accuracy schedule.
Key results
- On stable structure prediction on MP-20, CrysBFN reaches 64.35% match rate vs. 51.49% for DiffCSP and 61.39% for FlowMM.
- On ab initio generation for Carbon-24, CrysBFN reports 99.12% COV-P, up from 97.27% for DiffCSP.
- CrysBFN reaches 60.02% match rate with 10 network forward steps, versus DiffCSP's 51.49% at 2000 steps (~100x speedup).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use CrysBFN instead of DiffCSP | ab initio generation · Carbon-24 | COV-P (%): 97.27 -> 99.12 | Helped |
| use CrysBFN instead of DiffCSP | ab initio generation · Perov-5 | COV-P (%): 98.27 -> 98.63 | Helped |
| use CrysBFN instead of DiffCSP | ab initio generation · MP-20 | COV-P (%): 99.76 -> 99.79 | Helped |
| use CrysBFN instead of FlowMM | stable structure prediction · Perov-5 | Match rate: 53.15 -> 54.69 | Helped |
| use CrysBFN instead of DiffCSP | stable structure prediction · Perov-5 | RMSE: 0.0760 -> 0.0636 | Helped |
| use CrysBFN instead of FlowMM | stable structure prediction · MP-20 | Match rate: 61.39 -> 64.35 | Helped |
| use CrysBFN instead of FlowMM | stable structure prediction · MP-20 | RMSE: 0.0566 -> 0.0433 | Helped |
| use CrysBFN instead of FlowMM | stable structure prediction · MPTS-52 | Match rate: 17.54 -> 20.52 | Helped |
| use CrysBFN instead of FlowMM | stable structure prediction · MPTS-52 | RMSE: 0.1726 -> 0.1038 | Helped |
| add entropy conditioning instead of time conditioning (ablation) | CrysBFN · MP-20 stable structure prediction | Match rate (%): 52.16 -> 64.35 | Helped |
| use the searched linear-entropy sender accuracy schedule instead of the hand-designed roughly linear-entropy schedule (ablation) | CrysBFN · MP-20 stable structure prediction | Match rate (%): 49.76 -> 64.35 | Helped |
| use hyper-torus BFN instead of original continuous BFN (ablation) | CrysBFN · MP-20 stable structure prediction | Match rate (%): 6.17 -> 64.35 | Helped |
Surprising findings
- Replacing the entropy conditioning with time conditioning drops MP-20 match rate from 64.35% to 52.16%, showing time is a poor conditioning signal for the periodic flow.
- Using the original Gaussian-based BFN instead of the torus formulation collapses match rate to 6.17%.
Limitations
- Ab initio generation validity and COV-R are sometimes slightly below DiffCSP or CDVAE (e.g., MP-20 COV-R 99.09 vs 99.71 for DiffCSP; d_elem 0.1628 vs 0.083 for FlowMM).
- The accuracy schedule requires numerical binary search, though it is computed once and cached.
Experimental setup
Datasets: Perov-5 Carbon-24 MP-20 MPTS-52
Benchmarks: Perov-5 Carbon-24 MP-20 MPTS-52
Metrics: validity coverage (COV-R, COV-P) property EMD (density, energy, elements) match rate RMSE
Baselines: CDVAE Cond-DFC-VAE G-SchNet P-G-SchNet DiffCSP FlowMM
Builds on: Bayesian Flow Networks DiffCSP CSPNet
Closest papers by meaning
- LLM Meets Diffusion: A Hybrid Framework for Crystal Material Generation (NeurIPS 2025 · similarity 0.90)
- A Diffusion-Based Pre-training Framework for Crystal Property Prediction (AAAI 2024 · similarity 0.89)
- Scalable Diffusion for Materials Generation (ICLR 2024 · similarity 0.89)
- Kinetic Langevin Diffusion for Crystalline Materials Generation (ICML 2025 · similarity 0.89)
- Vector Field Oriented Diffusion Model for Crystal Material Generation (AAAI 2024 · similarity 0.89)
- WyckoffDiff -- A Generative Diffusion Model for Crystal Symmetry (ICML 2025 · similarity 0.88)
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