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Space Group Constrained Crystal Generation
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Proposes DiffCSP++, a diffusion model for crystal generation that enforces space group constraints by parameterizing the lattice via an O(3)-invariant logarithmic-space basis (per crystal family) and Wyckoff-position constraints on fractional coordinates.
Problem
Crystal generation models ignore space group symmetry, which governs crystal geometry and many properties; enforcing it in diffusion models is hard because space group constraints take diverse forms.
Approach
Reduce space group constraints to two tractable parts: restricting the lattice's logarithmic-space coefficients (fixed per crystal family) and restricting fractional coordinates to Wyckoff positions, then run a joint diffusion over these reduced variables with a projected denoiser that is O(3)-invariant.
Key results
- On MP-20 crystal structure prediction, DiffCSP++ (w/ GT space group) reaches 80.27% match rate vs 51.49% for DiffCSP; w/ CSPML templates it reaches 70.58%.
- On MPTS-52 CSP, DiffCSP++ (w/ CSPML) achieves 37.17% match rate vs 12.19% for DiffCSP.
- Ab initio generation on Perov-5 gives density distance 0.0661 vs 0.1110 for DiffCSP; on MP-20 element-distance is worse (0.3749 vs 0.3398 for DiffCSP).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use CSPML template conditions with DiffCSP++ instead of DiffCSP | crystal structure prediction · MP-20 | MR (%): 51.49 -> 70.58 | Helped |
| use CSPML template conditions with DiffCSP++ instead of DiffCSP | crystal structure prediction · MPTS-52 | RMSE: 0.1786 -> 0.0676 | Helped |
| use CSPML template conditions with DiffCSP++ instead of DiffCSP | crystal structure prediction · Perov-5 | RMSE: 0.0760 -> 0.0841 | Hurt |
| apply space-group-constrained DiffCSP++ instead of DiffCSP | ab initio generation · Perov-5 | dE: 0.0263 -> 0.0405 | Hurt |
| apply space-group-constrained DiffCSP++ instead of DiffCSP | ab initio generation · MP-20 | dE: 0.1247 -> 0.0574 | Helped |
| replace DiffCSP's lattice inner-product representation with coefficient vector k (ablation) | DiffCSP · MP-20 | MR (%): 51.49 -> 50.76 | Hurt |
| replace DiffCSP's lattice inner-product representation with coefficient vector k (ablation) | DiffCSP · MP-20 | RMSE: 0.0631 -> 0.0608 | Helped |
| use post-average rather than pre-average loss (ablation) | DiffCSP++ · MP-20 | MR (%): 78.75 -> 80.27 | Helped |
| use post-average rather than pre-average loss (ablation) | DiffCSP++ · MP-20 | RMSE: 0.0355 -> 0.0295 | Helped |
Surprising findings
- Replacing DiffCSP's lattice representation with the invariant coefficient vector k alone (DiffCSP-k) performs on par with original DiffCSP, so the gain comes mainly from the constraints rather than the representation.
Limitations
- Performance depends on the quality of space group and Wyckoff templates; a gap remains between CSPML-template and ground-truth space group conditions (70.58% vs 80.27% on MP-20).
- Ab initio generation is comparable rather than uniformly superior; some property metrics (e.g., element distance on MP-20, COV-P on Carbon-24) are worse than baselines.
Experimental setup
Datasets: Perov-5 Carbon-24 MP-20 MPTS-52
Benchmarks: Perov-5 Carbon-24 MP-20 MPTS-52
Metrics: match rate RMSE structural validity compositional validity COV-R COV-P Wasserstein density distance formation energy distance element number distance
Baselines: Random Search Bayesian Optimization Particle Swarm Optimization P-cG-SchNet CDVAE DiffCSP FTCP Cond-DFC-VAE G-SchNet P-G-SchNet CSPML
Builds on: DiffCSP CDVAE CSPML PyXtal
Closest papers by meaning
- Space Group Equivariant Crystal Diffusion (NeurIPS 2025 · similarity 0.91)
- Space Group Conditional Flow Matching (NeurIPS 2026 · similarity 0.88)
- Kinetic Langevin Diffusion for Crystalline Materials Generation (ICML 2025 · similarity 0.87)
- WyckoffDiff -- A Generative Diffusion Model for Crystal Symmetry (ICML 2025 · similarity 0.87)
- SymmCD: Symmetry-Preserving Crystal Generation with Diffusion Models (ICLR 2025 · similarity 0.87)
- Vector Field Oriented Diffusion Model for Crystal Material Generation (AAAI 2024 · similarity 0.87)
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