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Space Group Conditional Flow Matching

NeurIPS 2026 · Probabilistic and scientific modeling › Molecules and proteins · approach: Flow matching generative modeling

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Proposes Space Group Conditional Flow Matching (SGFM), a flow-matching crystal generator whose noise prior respects Wyckoff positions and whose vector field is space-group equivariant via an efficient group averaging, achieving SOTA on CSP and de novo generation benchmarks.

Problem

Generative models for inorganic crystals ignore space group and Wyckoff position constraints, so they produce many low-symmetry structures rather than the highly symmetric, stable crystals seen in nature.

Approach

Condition the whole flow on a space group and Wyckoff positions: a Wyckoff-conditioned symmetric noise prior plus a G x S_n equivariant vector field obtained by an efficient group averaging that needs only one network evaluation per step, since inputs are already G-symmetric.

Key results

• CSP on MP-20: match rate 82.74% vs 80.27% for DiffCSP++ (RMSE 0.0288 vs 0.0295).
• De novo generation on MP-20: S.U.N. 30.3% (E_hull<100 meV/atom) and 6.9% (E_hull<0), outperforming the listed baselines on S.U.N. metrics.
• Ablation on MP-20: equivariant model with Wyckoff-conditional prior reaches 82.74% MR vs 68.16% non-equivariant with the same prior and 64.49% with a uniform prior.

Surprising findings

• The Wyckoff-conditional prior alone helps a non-equivariant model only modestly (68.16% vs 64.49% uniform), while combining it with equivariance gives a large jump.

Limitations

• Requires space group and Wyckoff positions as conditioning inputs; CSP relies on ground truth or retrieved (CSPML) values.
• Stability evaluation uses DFT without guessed oxidation states, which may under-predict stability.
• Currently conditional only; unconditional generation of space group and Wyckoff positions is future work.

Experimental setup

Datasets: MP-20 MPTS-52 Alex-MP-20 Perov-5 Carbon-24

Benchmarks: MP-20 MPTS-52 Perov-5 Carbon-24 Alex-MP-20 CSPML evaluation S.U.N. de novo generation on MP-20

Metrics: match rate RMSE validity Wasserstein property divergence stable rate (E_hull) S.U.N.

Baselines: CDVAE FlowMM OMatG DiffCSP++ ADiT FlowLLM SymmCD WyFormer SGEquiDiff CSPML

Builds on: flow matching group averaging DiffCSP++ EGNN FlowMM

Closest papers by meaning

• Space Group Equivariant Crystal Diffusion (NeurIPS 2025 · similarity 0.91)
• Wyckoff Transformer: Generation of Symmetric Crystals (ICML 2025 · similarity 0.90)
• WyckoffDiff -- A Generative Diffusion Model for Crystal Symmetry (ICML 2025 · similarity 0.89)
• Space Group Constrained Crystal Generation (ICLR 2024 · similarity 0.88)
• Generating Symmetric Materials using Latent Flow Matching (NeurIPS 2026 · similarity 0.88)
• SymmCD: Symmetry-Preserving Crystal Generation with Diffusion Models (ICLR 2025 · similarity 0.88)