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SymmCD: Symmetry-Preserving Crystal Generation with Diffusion Models
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Proposes SymmCD, a diffusion model that generates crystals by jointly diffusing the asymmetric unit, Wyckoff-style site symmetry binary encodings, and space group, so generated crystals preserve chosen space group symmetry.
Problem
Existing crystal generative models either produce low-symmetry (mostly P1) crystals or replicate space-group distributions from the training data, failing to generate realistic and diverse crystal symmetries.
Approach
Represent crystals as asymmetric units with a space group, lattice vector k, representative coordinates, atom types and a novel binary (15 axes x 13 operations) site-symmetry encoding; diffuse over these components with continuous and discrete diffusion so that generation preserves the space group by construction and generalizes across groups.
Key results
- SymmCD (10 SGs) achieves 92.30% structural validity and 87.13% compositional validity on MP-20, with d_sg 0.469 vs 0.16 for DiffCSP++, and SymmCD (All SGs) reaches d_sg 0.164 matching DiffCSP++ (0.16).
- SymmCD generates 2794 unique templates out of 10,000 crystals on MP-20, the most of any method, with 1654 new templates not in training.
- After CHGNet relaxation, SymmCD (10 SGs) reaches 10.92% relaxed stable and 7.59% S.U.N., performing on par with baselines (DiffCSP 11.32% / 8.92%).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use asymmetric unit representation instead of a full conventional unit cell (ablation) | SymmCD · MP-20 | training epoch time: 52 sec. -> 27 sec. | Helped |
| Use asymmetric unit representation instead of a full conventional unit cell (ablation) | SymmCD · MP-20 | memory for batch size 512: 31 GB -> 3.6 GB | Helped |
| Use asymmetric unit representation instead of a full conventional unit cell (ablation) | SymmCD · MP-20 | maximum batch size: 512 -> 8192 | Helped |
| Restrict sampling to the 10 most common space groups (ablation) | SymmCD · MP-20 | stable after relaxation (%): 9.34% -> 10.92% | Helped |
| Restrict sampling to the 10 most common space groups (ablation) | SymmCD · MP-20 | recall (%): 99.58_{\pm .08 } -> 97.33_{\pm .59 } | Hurt |
| Add site symmetry encoding to asymmetric unit input (ablation) | DimeNet++ formation energy predictor · MP-20 test set | formation energy MAE: 0.0711 -> 0.0490 | Helped |
| Add site symmetry encoding to multigraph input (ablation) | DimeNet++ formation energy predictor · MP-20 test set | formation energy MAE: 0.0214 -> 0.0212 | No clear effect |
| Generate with SymmCD rather than DiffCSP++ | Crystal generation · MP-20 | new templates: 110 -> 1654 | Helped |
| Generate with SymmCD rather than DiffCSP++ | Crystal generation · MP-20 | structural validity (%): 99.44_{\pm 0.06 } -> 90.34_{\pm 3.39 } | Hurt |
| Generate with SymmCD rather than the baselines | Crystal generation · MP-20 | maximum force after DFT optimization | Helped |
Surprising findings
- SymmCD is more likely to generate structurally invalid crystals than baselines, possibly because it cannot easily see interatomic distances.
- Asymmetric-unit based training is about 1.8x faster per epoch and fits 8192 batch size vs 512 for the conventional cell, yet the paper reports DiffCSP++ and SymmCD produce the lowest relaxation displacements.
Limitations
- Relies on sampling a space group first, making crystal structure prediction given a composition more difficult.
- Structural validity is lower than baselines (e.g., 90.34% vs 99.97% for CDVAE), possibly due to lack of inter-atomic distance awareness.
- Only single crystals are considered; polycrystalline materials are not handled.
Experimental setup
Datasets: MP-20 MPTS-52
Benchmarks: MP-20 MPTS-52
Metrics: structural validity compositional validity coverage recall coverage precision d_rho d_E d_elem d_sg stable rate after CHGNet relaxation S.U.N. rate
Baselines: CDVAE DiffCSP DiffCSP++ FlowMM
Builds on: DiffCSP CDVAE D3PM discrete diffusion PyXtal
Closest papers by meaning
- WyckoffDiff -- A Generative Diffusion Model for Crystal Symmetry (ICML 2025 · similarity 0.94)
- Generating Symmetric Materials using Latent Flow Matching (NeurIPS 2026 · similarity 0.92)
- Kinetic Langevin Diffusion for Crystalline Materials Generation (ICML 2025 · similarity 0.92)
- Wyckoff Transformer: Generation of Symmetric Crystals (ICML 2025 · similarity 0.92)
- LLM Meets Diffusion: A Hybrid Framework for Crystal Material Generation (NeurIPS 2025 · similarity 0.90)
- Scalable Diffusion for Materials Generation (ICLR 2024 · similarity 0.90)
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