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Equivariant Neural Diffusion for Molecule Generation
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Proposes Equivariant Neural Diffusion (END), an E(3)-equivariant 3D molecule diffusion model with a learnable, data- and time-dependent forward process built on NFDM, evaluated on QM9 and GEOM-Drugs plus conditional QM9 generation.
Problem
Generating 3D molecules with diffusion models whose forward (noising) process is pre-specified and fixed, which may limit latent flexibility and make the reverse process harder to learn, while preserving E(3) symmetry.
Approach
Replace the fixed EDM forward process with a learnable transformation F(eps, t, x) = mu(x,t) + U(x,t) eps that is equivariant to rotations/reflections, so the learned forward and reverse processes share equivariance and the marginal stays invariant; combines NFDM's learnable forward with EDM's equivariant design.
Key results
- On QM9 with 1000 steps, END reaches 98.9% atom stability and 89.1% molecule stability versus EDM* 98.4% and 85.3%; END with 100 steps already beats EDM* at 1000 steps.
- On GEOM-Drugs, END at 1000 steps achieves 79.4% validity x connectivity versus 64.2% for EDM* and 45.8% for GeoLDM; with 100 steps END gets 73.7%.
- Conditional composition matching on QM9 rises to about 91% for cEND versus about 75.5% for cEDM* at 1000 steps; substructure Tanimoto similarity improves to 0.828 vs 0.673.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace the fixed forward process with a learnable forward process (ablation) | EDM* on QM9 · QM9, unconditional generation | molecule stability (%): 85.3 -> 89.1 | Helped |
| replace the fixed forward process with a learnable forward process (ablation) | EDM* on GEOM-Drugs · GEOM-Drugs, unconditional generation | validity × connectivity (%): 64.2 -> 79.4 | Helped |
| replace the fixed forward process with a learnable forward process (ablation) | EDM* on QM9 · QM9, unconditional generation | strain energy (kcal/mol): 11.3 -> 9.7 | Helped |
| replace the fixed forward process with a learnable forward process | cEDM* on QM9 · QM9 composition-conditioned | composition matching (%): 75.5 -> 91.0 | Helped |
| replace the fixed forward process with a learnable forward process | cEDM* on QM9 · QM9 substructure-conditioned | Tanimoto similarity: $.673_{\scriptstyle \pm.002}$ -> $.828_{\scriptstyle \pm.001}$ | Helped |
| fix the conditional marginal standard deviation (ablation) | END on QM9 · QM9, unconditional generation | molecule stability (%): $89.1_{\scriptstyle \pm.1}$ -> $89.2_{\scriptstyle \pm.3}$ | No clear effect |
| fix the conditional marginal standard deviation (ablation) | cEND on QM9 · QM9 composition-conditioned | composition matching (%): 91.0 -> 83.5 | Hurt |
| fix the conditional marginal standard deviation (ablation) | END on GEOM-Drugs · GEOM-Drugs, unconditional generation | strain energy (kcal/mol): 55.0 -> 71.0 | Hurt |
| learn modality-specific SNRs rather than a learnable forward process (ablation) | EDM* on QM9 · QM9, unconditional generation | atom-type TV ($10^{-2}$): $2.5_{\scriptstyle \pm.1}$ -> $2.1_{\scriptstyle \pm.1}$ | Helped |
| increase integration steps from 50 to 1000 | END on QM9 · QM9, unconditional generation | molecule stability (%): $84.6_{\scriptstyle \pm.1}$ -> $89.1_{\scriptstyle \pm.1}$ | Helped |
| increase integration steps from 50 to 1000 | END on GEOM-Drugs · GEOM-Drugs, unconditional generation | validity × connectivity (%): $66.0_{\scriptstyle \pm.4}$ -> $79.4_{\scriptstyle \pm.4}$ | Helped |
Surprising findings
- END with 100 sampling steps outperforms EDM* with 1000 steps on several metrics, suggesting learnable forward processes improve sampling efficiency even though this was not explicitly optimized.
Limitations
- Training and sampling are slower per step than the fixed-forward EDM baseline (about 2.5x and 3x).
- Operates on fully connected graphs, limiting scaling to very large molecules, and models categorical atom types via continuous relaxation.
- Evaluated only on organic molecules with computational proxy metrics; no QM validation of generated molecules.
Experimental setup
Datasets: QM9 GEOM-Drugs
Benchmarks: QM9 GEOM-Drugs
Metrics: atom stability molecule stability validity uniqueness connectivity total variation strain energy Tanimoto similarity composition matching SA
Baselines: EDM EDM-Bridge GeoLDM GeoBFN EEGSDE cEDM
Builds on: Equivariant Diffusion Models (EDM) Neural Flow Diffusion Models (NFDM) EDM-Bridge GeoLDM
Closest papers by meaning
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