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FreeCG: Free the Design Space of Clebsch-Gordan Transform for Machine Learning Force Fields
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Proposes FreeCG, which applies Clebsch-Gordan transforms on permutation-invariant "abstract edges" to free the CG design space, yielding group CG transform with sparse paths, abstract edge shuffling and an attention enhancer for molecular force field prediction.
Problem
Permutation-equivariance requirements in equivariant graph neural network force fields restrict the design space of Clebsch-Gordan (CG) transforms, which limits expressivity and makes the expensive CG computation scale poorly with the number of neighbors.
Approach
Build permutation-invariant abstract edges by attention-weighted aggregation of real edge features (ViSNet-style); since any function of invariant inputs is invariant regardless of its design, CG transforms applied to abstract edges can be designed freely (per-edge weights, groups, sparse paths, shuffling) while preserving O(3) equivariance.
Key results
- On MD17 force prediction, FreeCG reports best MAE on all seven molecules, e.g. aspirin 0.122 vs 0.145 for QuinNet (about 15% lower).
- On rMD17 force prediction, FreeCG achieves the best force MAE on most molecules, e.g. aspirin 0.1212 vs 0.1429 for QuinNet and salicylic acid 0.0648 vs 0.0771.
- On QM9, FreeCG is best on 7 of 12 targets, e.g. alpha 38.2 vs 41.1 (ViSNet) and C_v 20.4 vs 23 (ViSNet).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace ViSNet with FreeCG | ViSNet · MD17 force prediction (most molecules) | Force MAE: FreeCG outperforms others in all force prediction tasks | Helped |
| Replace ViSNet with FreeCG | ViSNet · rMD17 force prediction (most molecules) | Force MAE | Mixed |
| Replace ViSNet with FreeCG | ViSNet · MD22 force prediction | Force MAE | Mixed |
| Replace ViSNet with FreeCG | ViSNet · QM9 | MAE | Mixed |
| Add Group CG transform (ablation) | ViSNet on MD17 Aspirin · MD17 Aspirin | Force MAE: 0.155 -> 0.144 | Helped |
| Use 32 rather than 8 groups for Group CG transform (ablation) | ViSNet on MD17 Aspirin · MD17 Aspirin | Force MAE: 0.144 -> 0.129 | Helped |
| Add abstract-edge shuffling (1-group shuffle) (ablation) | Group CG transform, 32 groups · MD17 Aspirin | Force MAE: 0.129 -> 0.128 | Helped |
| Use 0.5-group rather than 1-group shuffle (ablation) | Group CG transform, 32 groups · MD17 Aspirin | Force MAE: 0.128 -> 0.128 | No clear effect |
| Use 1.5-group rather than 1-group shuffle (ablation) | Group CG transform, 32 groups · MD17 Aspirin | Force MAE: 0.128 -> 0.125 | Helped |
| Add the attention enhancer (ablation) | Group CG transform with the selected abstract-edge shuffle · MD17 Aspirin | Force MAE: 0.125 -> 0.122 | Helped |
Surprising findings
- FreeCG is worse than ViSNet on several rMD17 energy and some QM9 targets, such as <R^2> (82.1 vs 29.8 for ViSNet) and toluene energy in rMD17 (0.0334 vs 0.0059 for ViSNet).
Limitations
- Energy prediction is not uniformly best; FreeCG underperforms ViSNet on several energy and property targets.
- Code is only promised ('will be publicly available'), not released at submission.
- Hyperparameters such as group number and shuffle strides are chosen empirically.
Experimental setup
Datasets: MD17 rMD17 MD22 QM9 AIMD-Chig water-1k LiPS
Benchmarks: MD17 rMD17 MD22 QM9 water-1k LiPS AIMD-Chig
Metrics: MAE inference speed memory usage RDF recovery
Baselines: ViSNet NequIP Allegro MACE QuinNet SO3KRATES ET GemNet SphereNet DimeNet
Builds on: ViSNet NequIP QuinNet ShuffleNet PaiNN
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