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SymDiff: Equivariant Diffusion via Stochastic Symmetrisation
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Proposes SymDiff, which builds stochastically E(3)-equivariant diffusion models for molecules by stochastically symmetrising non-equivariant (S_N-equivariant) DiT denoisers with a learned Orth(3) distribution, avoiding intrinsically equivariant networks.
Problem
Constructing equivariant diffusion models for N-body systems (molecules) without relying on intrinsically equivariant neural networks, which are costly, hard to implement, and optimise poorly.
Approach
Apply stochastic symmetrisation (from Markov categories theory) to Markov kernels: sample a random orthogonal transform g from a learned kernel gamma(dg|z_t), apply the S_N-equivariant denoiser to g^{-1}z_t, and map the output back by g. This yields stochastic E(3) equivariance with a single pass and no bias, generalising data augmentation (which is recovered when gamma is Haar measure), and allow
Key results
- On QM9, SymDiff (31.2M params) reaches NLL -129.35 and molecular stability 87.49%, versus EDM at -110.70 NLL and 82.00% molecular stability.
- SymDiff* (115.6M backbone) achieves the best atom stability (98.92%) and validity (96.36%) on QM9.
- On GEOM-Drugs, SymDiff achieves NLL -301.21 and atom stability 86.16% versus EDM's -137.1 NLL and 81.3% atom stability.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace EDM reverse kernels with SymDiff | EDM · QM9 | validity (%): 91.90$\spm{0.50}$ -> {95.75} $\spm {0.10}$ | Helped |
| replace EDM reverse kernels with SymDiff | EDM · GEOM-Drugs | atom stability (%): 81.3 -> 86.16$\spm{0.05}$ | Helped |
| increase SymDiff model size from 31.2M to 117.8M parameters (ablation) | SymDiff · QM9 | validity (%): {95.75} $\spm {0.10}$ -> 96.36$\spm{0.27}$ | Helped |
| reduce SymDiff model size to 23.5M parameters (ablation) | SymDiff · QM9 | molecular stability (%): 87.49 $\spm{0.23}$ -> 83.51$\spm{0.24}$ | Hurt |
| reduce SymDiff model size to 13.5M parameters (ablation) | SymDiff · QM9 | molecular stability (%): 87.49 $\spm{0.23}$ -> 71.25 $\spm{0.50}$ | Hurt |
| increase EDM model size to 9.5M parameters (ablation) | EDM · QM9 | molecular stability (%): 82.00 $\spm{0.40}$ -> 84.63$\spm{0.16}$ | Helped |
| increase EDM model size to 12.4M parameters (ablation) | EDM · QM9 | molecular stability (%): 82.00 $\spm{0.40}$ -> 85.68$\spm{0.83}$ | Helped |
| use fixed Haar measure instead of learning the symmetrisation component (ablation) | SymDiff · QM9 | molecular stability (%): 87.49 $\spm{0.23}$ -> 85.51$\spm{0.18}$ | Hurt |
| apply random orthogonal data augmentation instead of learned symmetrisation (ablation) | DiT · QM9 | molecular stability (%): 81.03$\spm{0.25}$ -> 85.85$\spm{0.24}$ | Helped |
| initialize with pretrained DiT and finetune both SymDiff components (ablation) | DiT · QM9 | molecular stability (%): 81.03$\spm{0.25}$ -> 83.27 $\spm{0.39}$ | Helped |
| freeze pretrained DiT and finetune only the symmetrisation component (ablation) | DiT · QM9 | molecular stability (%): 81.03$\spm{0.25}$ -> 55.02$\spm{0.38}$ | Hurt |
| use SymDiff rather than intrinsically equivariant baselines | molecular generation models · QM9 | seconds per epoch: 27.20 vs 88.80 | Helped |
Surprising findings
- Data augmentation with a DiT backbone substantially improves over the plain DiT on all QM9 metrics.
Limitations
- Results largely limited to E(3)-equivariant molecular generation on QM9 and GEOM-Drugs.
- Architecture choices were minimally tuned.
Experimental setup
Baselines: EDM GeoLDM END MUDiff DiT DiT-Aug SymDiff-H
Builds on: EDM Markov categories DiT Set Transformer
Closest papers by meaning
- Equivariant Neural Diffusion for Molecule Generation (NeurIPS 2024 · similarity 0.90)
- SymDrift: One-Shot Generative Modeling under Symmetries (NeurIPS 2026 · similarity 0.90)
- Scalable Non-Equivariant 3D Molecule Generation via Rotational Alignment (ICML 2025 · similarity 0.88)
- The Superposition of Diffusion Models Using the Itô Density Estimator (ICLR 2025 · similarity 0.87)
- Rao-Blackwell Gradient Estimators for Equivariant Denoising Diffusion (NeurIPS 2025 · similarity 0.87)
- Derivative-Free Guidance in Continuous and Discrete Diffusion Models with Soft Value-based Decoding (NeurIPS 2025 · similarity 0.87)
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