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Symphony: Symmetry-Equivariant Point-Centered Spherical Harmonics for 3D Molecule Generation
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Symphony is an E(3)-equivariant autoregressive molecule generator that builds molecules fragment by fragment, predicting atom placement as spherical-harmonic signals from a single focus atom, and approaches diffusion-model quality on QM9.
Problem
Autoregressive 3D molecule generation has relied on rotation-invariant features, which forces triangulation with auxiliary tokens and non-smooth dependence on the fragment; diffusion models are expensive and slow to sample, and do not easily allow fragment completion.
Approach
Use message passing with higher-degree E(3)-equivariant features (E3SchNet for focus/species, NequIP for position) and parametrize the distribution of the next atom's position as spherical-harmonic logits over radial shells around a focus atom, with multiple channels to increase angular frequency capacity.
Key results
- Symphony reaches 83.50% validity via xyz2mol vs. 86.74% for EDM, 74.97% for G-SchNet and 26.92% for G-SphereNet on QM9.
- Symphony attains the lowest bispectrum MMD on several local environments, and the lowest JSD for local environment counts (0.0039 vs. 0.0057 for EDM).
- Symphony samples at 0.293 s/molecule versus 0.930 s/molecule for EDM on a single RTX A5000, though slower than G-SchNet (0.011) and G-SphereNet (0.006).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Symphony instead of G-SchNet | QM9 molecule generation · QM9 generated molecules, xyz2mol validity | validity (%): 74.97 -> 83.50 | Helped |
| Use Symphony instead of EDM | QM9 molecule generation · QM9 generated molecules, xyz2mol validity | validity (%): 86.74 -> 83.50 | Hurt |
| Use Symphony instead of EDM | QM9 molecule generation · QM9 generated molecules, Lookup Table validity | validity (%): 90.77 -> 68.11 | Hurt |
| Use Symphony instead of EDM | QM9 molecule generation · QM9 generated molecules, reasonable internal energy | passing molecules (%): 94.88 -> 95.65 | Helped |
| Use Symphony instead of EDM | QM9 molecule generation · QM9 generated molecules, double bond flatness | passing molecules (%): 98.58 -> 99.07 | Helped |
| Use Symphony instead of EDM | QM9 molecule generation · QM9 bispectrum local environments | MMD | Mixed |
| Train Symphony for 800K or 1000K rather than 500K steps | Symphony · QM9 fragment completion | valid completion rate (%): 98.53 -> 96.65; 98.53 -> 95.57 | Hurt |
| Use Symphony instead of EDM | QM9 molecule generation · QM9 molecule generation | inference time (seconds/molecule): 0.930 sec/mol -> 0.293 seconds/molecule | Helped |
| Use Symphony instead of G-SchNet | QM9 molecule generation · QM9 molecule generation | inference time (seconds/molecule): 0.011 sec/mol -> 0.293 seconds/molecule | Hurt |
| Increase focus embedder E3SchNet lmax from 1 to 2 (ablation) | Symphony · QM9 generated molecules | xyz2mol validity | No clear effect |
| Increase position embedder NequIP lmax from 1 to 2 (ablation) | Symphony · QM9 generated molecules | xyz2mol validity | Helped |
| Increase position embedder NequIP lmax above 2 (ablation) | Symphony · QM9 generated molecules | xyz2mol validity | Helped |
Surprising findings
- Symphony's valid completion rate decreased as training progressed (98.53% at 500K steps to 95.57% at 1000K steps), with the cause left unclear.
- EDM with distance cutoffs performs quite poorly, suggesting fully-connected graphs matter for diffusion models.
Limitations
- Coarse radial discretization (64 bins) makes bond length distributions wider than QM9, hurting validity under lookup-table bond assignment and MMD of bond lengths.
- Higher-degree tensor products increase computational cost, making sampling slower than other autoregressive models.
- Evaluated only on QM9 small molecules (up to 30 atoms).
Experimental setup
Datasets: QM9
Benchmarks: QM9 PoseBusters
Metrics: validity uniqueness MMD of bond lengths MMD of bispectra Jensen-Shannon divergence valid completion rate inference seconds per molecule
Baselines: EDM G-SchNet G-SphereNet
Builds on: G-SchNet G-SphereNet NequIP SchNet EDM
Closest papers by meaning
- Scalable Non-Equivariant 3D Molecule Generation via Rotational Alignment (ICML 2025 · similarity 0.90)
- 3D Molecule Generation from Rigid Motifs via $\mathrm{SE}(3)$ Flows (NeurIPS 2026 · similarity 0.87)
- Navigating the Design Space of Equivariant Diffusion-Based Generative Models for De Novo 3D Molecule Generation (ICLR 2024 · similarity 0.87)
- Equivariant Neural Diffusion for Molecule Generation (NeurIPS 2024 · similarity 0.87)
- SymDrift: One-Shot Generative Modeling under Symmetries (NeurIPS 2026 · similarity 0.87)
- NEAT: Neighborhood-Guided, Efficient, Autoregressive Set Transformer for 3D Molecular Generation (NeurIPS 2026 · similarity 0.86)
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