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Stiefel Flow Matching for Moment-Constrained Structure Elucidation
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Proposes Stiefel Flow Matching, a Riemannian flow matching model on the Stiefel manifold that enforces exact moment-of-inertia constraints for 3D molecular structure elucidation from formula and moments, with approximate equivariant optimal transport.
Problem
Predicting a molecule's all-atom 3D structure from its molecular formula and moments of inertia (measurable by rotational spectroscopy), where prior Euclidean diffusion models only softly satisfy the moment constraints.
Approach
Embeds the set of n-atom point clouds with fixed planar moments of inertia into the Stiefel manifold St(n,4), so that Riemannian flow matching on the Stiefel manifold generates structures that satisfy the moments exactly, combined with approximate equivariant optimal transport (permutation and reflection alignment) to shorten flows.
Key results
- On QM9, Stiefel FM achieves 15.17% success at 0.25 Å RMSD versus 13.65% for KREED-XL, using 200 NFE versus 1000 for diffusion baselines.
- On GEOM, Stiefel FM-OT (filter) reaches 3.94% success at 0.25 Å, the highest among methods, but with a mediocre stability score.
- Equivariant OT shortens average QM9 generation trajectory length from 1.696 to 1.547.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use Stiefel Flow Matching instead of Euclidean diffusion | Structure elucidation models · QM9 test | % < RMSD 0.25 Å: 13.67±0.30 -> 15.17±0.31 | Helped |
| add equivariant optimal transport (ablation) | Stiefel FM · QM9 test | % < RMSD 0.25 Å: 15.17±0.31 -> 13.99±0.30 | Hurt |
| add equivariant optimal transport (ablation) | Stiefel FM · GEOM test | % < RMSD 0.25 Å: 2.17±0.09 -> 2.44±0.09 | Helped |
| filter generated samples to retain up to K=10 valid samples (ablation) | Stiefel FM · GEOM test | % < RMSD 0.25 Å: 2.17±0.09 -> 3.57±0.11 | Helped |
| add equivariant optimal transport and filter generated samples to retain up to K=10 valid samples (ablation) | Stiefel FM · GEOM test | % < RMSD 0.25 Å: 2.17±0.09 -> 3.94±0.11 | Helped |
| enforce exact moment constraints through Stiefel manifold generation | Stiefel FM · QM9 test | Error: 5.18 -> 0.00 | Helped |
Surprising findings
- Optimal transport slightly reduces Stiefel FM success rate on QM9 while improving it on GEOM.
- Projecting KREED-XL samples onto the feasible manifold does not change success rate, though it removes moment violations.
Limitations
- On GEOM, Stiefel FM generates few valid structures (about 39%) and has low accuracy, suggesting underfitting on large molecules.
- Requires the molecular formula as input and cannot handle exactly planar or linear molecules.
- Success is measured against a single ground-truth structure, so the evaluation relies on many generated samples.
Experimental setup
Datasets: QM9 GEOM-Drugs
Benchmarks: QM9 GEOM-Drugs
Metrics: success rate at RMSD < 0.25 Å success rate at RMSD < 0.10 Å moment error validity stability (xtb gradient norm) diversity NFE
Baselines: KREED KREED-XL KREED-XL-DPS KREED-XL-proj Stiefel Random
Builds on: Riemannian flow matching KREED EDM Diffusion Posterior Sampling
Closest papers by meaning
- Energy-Based Flow Matching for Generating 3D Molecular Structure (ICML 2025 · similarity 0.85)
- Accelerating 3D Molecule Generation via Jointly Geometric Optimal Transport (ICLR 2025 · similarity 0.85)
- Symphony: Symmetry-Equivariant Point-Centered Spherical Harmonics for 3D Molecule Generation (ICLR 2024 · similarity 0.85)
- DiffMS: Diffusion Generation of Molecules Conditioned on Mass Spectra (ICML 2025 · similarity 0.84)
- SE(3)-Stochastic Flow Matching for Protein Backbone Generation (ICLR 2024 · similarity 0.84)
- Scalable Non-Equivariant 3D Molecule Generation via Rotational Alignment (ICML 2025 · similarity 0.84)
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