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Weisfeiler Leman for Euclidean Equivariant Machine Learning
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Shows analytic PPGN with low-dimensional features can uniformly simulate 2-WL on point clouds and position-velocity pairs, proves equivariant universality via a simple pooling layer, and introduces WeLNet which achieves SOTA on N-body and GEOM-QM9 conformation generation (MAT).
Problem
Existing 2-WL-based GNNs for point clouds only separate graph pairs non-uniformly with prohibitively high complexity, cover only invariant tasks, and do not handle joint position-velocity inputs or permutation-rotation-translation equivariant functions.
Approach
Shows that analytic non-polynomial PPGN with feature dimension 2d+1 uniformly separates 2-WL-separable families with O(n^omega) complexity; extends 2-WL completeness to position-velocity pairs; and adds a simple equivariant pooling layer on PPGN features that yields a universal equivariant architecture (WeLNet).
Key results
- WeLNet achieves test MSE 0.0036 on the N-body task, improving over the previous SOTA CGENN (0.0039).
- On GEOM-QM9 WeLNet obtains MAT mean 0.1614 Å, over 15% better than UniMol (0.1907), while COV is competitive (92.66% vs UniMol 97.00%).
- With a single neuron per node-pair, analytic activations (softplus, LeakyELU) separate all 600 EXP graph pairs whereas ReLU does not.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use WeLNet instead of CGENN | N-body dynamics prediction · N-body dynamics | MSE: 0.0039 $\pm$ 0.0001 -> 0.0036 $\pm$ 0.0002 | Helped |
| Use WeLNet instead of MC-EGNN | N-body with Gravity · N-body with Gravity | MSE: 0.0073$\pm$0.0002 -> 0.0054 $\pm$ 0.0001 | Helped |
| Use WeLNet instead of MC-EGNN | N-body with Lorentz-like force · N-body with Lorentz-like force | MSE: 0.0240$\pm$ 0.0010 -> 0.0238$\pm$0.0002 | Helped |
| Use WeLNet instead of UniMol | GEOM-QM9 conformation generation · GEOM-QM9 | COV mean (%): 97.00 -> 92.66 | Hurt |
| Use WeLNet instead of UniMol | GEOM-QM9 conformation generation · GEOM-QM9 | COV median (%): 100.00 -> 95.29 | Hurt |
| Use WeLNet instead of UniMol | GEOM-QM9 conformation generation · GEOM-QM9 | MAT mean (Å): 0.1907 -> 0.1614 | Helped |
| Use WeLNet instead of UniMol | GEOM-QM9 conformation generation · GEOM-QM9 | MAT median (Å): 0.1754 -> 0.1566 | Helped |
| Use an analytic activation rather than ReLU in single-feature PPGN (ablation) | PPGN graph-pair separation · EXP | separation | Helped |
| Increase ReLU PPGN width from one to 15 features per node-pair (ablation) | PPGN graph-pair separation · EXP | separation | Helped |
Surprising findings
- ReLU and LeakyReLU fail to separate all EXP pairs with one feature per node-pair, while analytic activations succeed, and ReLU needs 15 features to achieve perfect separation.
- WeLNet's COV on GEOM-QM9 is below UniMol's, so it is not best on every metric.
Limitations
- Universality and completeness results hold only for complete distance matrices, not for geometric graphs with distance cutoffs.
- Provable completeness requires O(n^omega) cost per PPGN block, a non-trivial computational cost.
- Conformation generation COV is below UniMol's best.
Experimental setup
Datasets: N-body charged particles simulation (3,000 train trajectories) EXP GEOM-QM9
Benchmarks: N-body dynamics (charged particles) N-body with gravity and Lorentz force GEOM-QM9 EXP
Metrics: MSE COV MAT separation gap
Baselines: EGNN ClofNet SEGNN CGENN Transformer-PS MC-EGNN UniMol GeoDiff DMCG
Builds on: PPGN 2-WL EGNN ClofNet ConfGF
Closest papers by meaning
- Universally Invariant Learning in Equivariant GNNs (NeurIPS 2025 · similarity 0.88)
- On the Generalization of Equivariant Graph Neural Networks (ICML 2024 · similarity 0.88)
- Complete Neural Networks for Complete Euclidean Graphs (AAAI 2024 · similarity 0.88)
- Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks? (NeurIPS 2024 · similarity 0.87)
- A Characterization Theorem for Equivariant Networks with Point-wise Activations (ICLR 2024 · similarity 0.86)
- On Universality Classes of Equivariant Networks (NeurIPS 2025 · similarity 0.86)
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