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E(3)-equivariant models cannot learn chirality: Field-based molecular generation
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Shows that E(3)-invariant point-cloud diffusion models cannot distinguish enantiomers and proposes Field-based Molecule Generation (FMG), a 3D U-Net diffusion model over atom/bond RBF fields that uses principal-component reference orientation to capture chirality while staying competitive on QM9 and GEOM-Drugs.
Problem
Point-cloud molecular generative models built on E(3)-invariant GNNs assign equal probability to enantiomers, so they cannot model chirality, which matters for drug potency and safety.
Approach
Represent molecules as continuous 3D atom and bond RBF fields, generate them with a 3D U-Net DDPM, and replace rotational symmetry constraints with canonical principal-component reference rotations so reflections (enantiomers) are distinguishable.
Key results
- FMG achieves 94.7% atom neutrality and 16.5% molecule neutrality on GEOM-Drugs versus EDM's 81.0% and 0.5%, and 99.4% atom / 91.5% molecule neutrality on QM9.
- On the chirality experiment, EDM produces equal probability for det R and -det R (enantiomer invariance), while FMG matches the data enantiomer distribution.
- FMG is competitive but not uniformly best on conformation metrics; on QM9 CE_xTB it scores 0.71 versus JODO's 0.22, and on GEOM-Drugs its BL W1 is 2.74 vs EDM's 8.84.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use reference rotation alignment rather than randomly rotated training fields (ablation) | FMG · QM9 | molecule neutrality (%): 82.6 -> 91.5 | Helped |
| add atom-count classifier-free guidance (ablation) | FMG · QM9 | TVa: 1.02 -> 0.52 | Helped |
| add atom-count classifier-free guidance (ablation) | FMG · QM9 | CE_xTB W1: 0.98 -> 0.71 | Helped |
| increase atom-count classifier-free guidance from beta 2 to beta 5 (ablation) | FMG · QM9 | TVa: 0.52 -> 0.48 | Helped |
| increase atom-count classifier-free guidance from beta 2 to beta 5 (ablation) | FMG · QM9 | TVb: 0.95 -> 1.02 | Hurt |
| extract bonds from pairwise atom distances instead of bond fields (ablation) | FMG · QM9 | molecule neutrality (%): 91.5 -> 90.9 | Hurt |
| extract bonds from pairwise atom distances instead of bond fields (ablation) | FMG · QM9 | validity (%): 98.8 -> 97.7 | Hurt |
| use atom-count guidance for conditional generation (ablation) | FMG · QM9 conditional generation, alpha | L1: 3.92 -> 3.55 | Helped |
| use atom-count guidance for conditional generation (ablation) | FMG · QM9 conditional generation, delta epsilon | molecule neutrality (%): 83.0 -> 84.5 | Helped |
| use fields instead of point-cloud E(3)-equivariant generation | molecular generation · synthetic chiral molecule distribution | enantiomer distribution match | Helped |
Surprising findings
- Removing rotation alignment (FMG_aug with random rotations) drops molecule neutrality to 82.6% on QM9, suggesting the reference orientation is the main driver of neutrality gains.
- Atom count conditioning trades TV_a for molecule neutrality; FMG_beta=0 gets higher molecule neutrality (92.2%) but worse TV_a (1.02).
Limitations
- Field tensor dimensions are large, and computational feasibility depends on molecule size and grid resolution.
- Reference rotations restrict generated molecules to the principal-component frame, so the model is not rotation-invariant.
- Molecule graph extraction from fields uses post-generation peak picking and optimization, complicating direct comparison with pure generative methods.
Experimental setup
Datasets: QM9 GEOM-Drugs
Benchmarks: QM9 GEOM-Drugs
Metrics: atom neutrality molecule neutrality validity MST TV_a TV_b bond length W1 bond angle W1 CE_MMFF W1 CE_xTB W1
Baselines: EDM MiDi JODO GeoLDM VoxMol MolDiff GDM-AUG EDM-Bridge
Builds on: DDPM EDM classifier-free guidance MiDi JODO
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