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Space Group Equivariant Crystal Diffusion
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Proposes SGEquiDiff, a space-group-constrained crystal generative model combining telescoping discrete lattice sampling, transformer-based Wyckoff/element sampling, and space group equivariant diffusion of atomic coordinates with explicit invariant likelihoods.
Problem
Crystal generative models typically ignore space group and Wyckoff position constraints, assigning zero probability to special positions and producing likelihoods that are not space group invariant, which limits generation of realistic high-symmetry crystals.
Approach
Factorizes crystal generation into space-group-invariant conditional samplers: telescoping discrete lattice sampling, autoregressive Wyckoff/element transformer, and diffusion of fractional coordinates using a Space Group Wrapped Normal distribution whose score is space group equivariant, with a theorem that equivariant vector fields are tangent to Wyckoff positions.
Key results
- On MP20, SGEquiDiff achieves the highest S.U.N. rate (24.4%) and a U.N. template rate of 18.6%, with 5.5M parameters versus MatterGen's 44.6M.
- On MPTS-52, SGEquiDiff attains the highest U.N. template rate (38.7%) and composition validity (79.83%).
- In crystal structure prediction on MP-20, SGEquiDiff+LDiff with CSPML reaches 69.42% match rate versus DiffCSP++ (+CSPML) at 70.58%.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace autoregressive lattice sampling with joint lattice and coordinate diffusion (ablation) | SGEquiDiff · MP-20 | Wasserstein distance for number of elements: 0.2092 -> 0.1292 | Helped |
| Replace autoregressive lattice sampling with joint lattice and coordinate diffusion (ablation) | SGEquiDiff · MP-20 | CMD of structural fingerprints: 0.1718 -> 0.1549 | Helped |
| Replace autoregressive lattice sampling with joint lattice and coordinate diffusion (ablation) | SGEquiDiff · MP-20 | structure validity: 99.25 -> 97.02 | Hurt |
| Replace autoregressive lattice sampling with joint lattice and coordinate diffusion (ablation) | SGEquiDiff · MP-20 | S.U.N. rate: 25.80 -> 13.65 | Hurt |
| Replace SGEquiDiff space-group and Wyckoff features with SymmCD features (ablation) | SGEquiDiff · MP-20 | structure validity: 99.25 -> 98.55 | Hurt |
| Replace SGEquiDiff space-group and Wyckoff features with SymmCD features (ablation) | SGEquiDiff · MP-20 | U.N. structure rate: 80.0 -> 84.9 | Helped |
| Replace the FAENet-inspired GNN with CSPNet (ablation) | SGEquiDiff · MP-20 | CMD of structural fingerprints: 0.1718 -> 0.1823 | Hurt |
| Increase inference-time categorical temperature from T=1.0 to T=2.0 (ablation) | SGEquiDiff · MP-20 | U.N. structure rate: 80.0 -> 95.0 | Helped |
| Increase inference-time categorical temperature from T=1.0 to T=2.0 (ablation) | SGEquiDiff · MP-20 | composition validity: 86.16 -> 81.08 | Hurt |
| Use SGEquiDiff +LDiff with CSPML template selection and element substitution | SGEquiDiff · MP-20 crystal structure prediction | match rate: 69.42% MR versus 70.58% for DiffCSP++ (+CSPML) | Mixed |
Surprising findings
- Replacing the autoregressive lattice sampler with joint DDPM lattice diffusion (+LDiff) raised S.U.N. rate to 25.80% on MP-20 with much higher validity, though it produced negative-determinant lattices in about 1% of samples.
- Raising inference-time temperature increased U.N. template rate to 57.1% at T=3.0 at the cost of worse distributional distances.
Limitations
- S.U.N. rates computed on only 1,000 valid crystals per model, adding unknown variance.
- Training data space group assignments used nonzero tolerances that may have symmetrized some materials incorrectly.
- Does not address other symmetry groups (magnetic, spin, layer) or synthesis guidance.
Experimental setup
Datasets: MP20 MPTS52
Benchmarks: MP20 MPTS52
Metrics: validity U.N. rate Wasserstein distance JSD CMD diversity S.U.N. rate match rate RMSE
Baselines: CDVAE DiffCSP DiffCSP++ SymmCD MatterGen FlowMM
Builds on: DiffCSP DiffCSP++ SymmCD CrystalFormer FAENet
Closest papers by meaning
- Space Group Conditional Flow Matching (NeurIPS 2026 · similarity 0.91)
- Space Group Constrained Crystal Generation (ICLR 2024 · similarity 0.91)
- WyckoffDiff -- A Generative Diffusion Model for Crystal Symmetry (ICML 2025 · similarity 0.90)
- SymmCD: Symmetry-Preserving Crystal Generation with Diffusion Models (ICLR 2025 · similarity 0.89)
- A Space Group Symmetry Informed Network for O(3) Equivariant Crystal Tensor Prediction (ICML 2024 · similarity 0.89)
- Wyckoff Transformer: Generation of Symmetric Crystals (ICML 2025 · similarity 0.89)
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