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Universally Invariant Learning in Equivariant GNNs
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Proposes Uni-EGNN, a dynamic-weight framework that makes equivariant GNNs complete by combining a complete scalar canonical form of geometric graphs with a full-rank steerable basis, yielding single-layer complete EGNN/TFN variants.
Problem
Achieving completeness (universal approximation over equivariant functions) for equivariant GNNs usually requires deep stacks, high body order, or high steerable degrees, which are computationally expensive and lack polynomial-time solutions.
Approach
Decompose equivariant function approximation into a complete scalar function (canonical form of the geometric graph, built from four virtual reference nodes generated by an E(3)-equivariant network after node coloring) and a full-rank steerable basis set; the output is a input-dependent weighted sum of basis features, so a single layer suffices.
Key results
- On the 5-body system, EGNN_cpl-local with half hidden dimension reaches MSE 0.450e-2 at 4 layers, versus 0.712e-2 for EGNN at 4 layers.
- On the tetrahedron center prediction task, EGNN_cpl-global reaches incenter MSE 0.025e-2 at 1 layer, versus 32.54e-2 for EGNN at 1 layer, showing single-layer completeness.
- Models with canonical form reach 100% accuracy on completeness tests regardless of whether the base model is complete.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add the canonical-form dynamic method to EGNN | EGNN · Completeness test: 2-body and 3-body tasks | accuracy | Helped |
| use tensor-product coloring instead of distance-to-center coloring (ablation) | Basic model with determinant-based chirality identification · Chirality test | graph distinction | Helped |
| add node coloring (ablation) | EGNN · Single-layer completeness test | loss | Helped |
| add the dynamic method | EGNN and TFN · Physical-system experiments | prediction error | Helped |
| use local rather than global virtual nodes (ablation) | EGNN_cpl · Water-3D-mini, 1-layer | MSE loss (× 10^-4): 4.368 -> 3.611 | Helped |
| use local rather than global virtual nodes (ablation) | EGNN_cpl · 100-body system | MSE loss (× 10^-2): 0.98 -> 0.73 | Helped |
| use half the standard node-embedding hidden dimension (ablation) | EGNN_cpl-local · 5-body system, 1-layer | MSE loss (× 10^-2): 0.768 -> 0.703 | Helped |
| use half the standard node-embedding hidden dimension (ablation) | EGNN_cpl-local · Tetrahedron Monge Point, 1-layer | MSE: 0.460 -> 0.460 | Hurt |
| use local rather than global virtual nodes (ablation) | EGNN_cpl · MD17 Benzene | Prediction error (× 10^-2): 58.24 -> 44.90 | Helped |
| use local rather than global virtual nodes (ablation) | EGNN_cpl · MD17 Toluene | Prediction error (× 10^-2): 10.37 -> 10.21 | Helped |
| increase the number of layers (ablation) | EGNN · 5-body system | MSE loss (× 10^-2): 4.214 -> 0.712 | Helped |
| increase the number of layers (ablation) | EGNN_cpl-local · Water-3D-mini | MSE loss (× 10^-4): 3.611 -> 2.495 | Helped |
Surprising findings
- Single-layer EGNN with canonical form matches or beats multi-layer baselines on several tasks, suggesting scalar-based models can be complete without deep stacking.
- Scalar models with spherical-scalarization (e.g., HEGNN) are conjectured possibly unable to build full-rank bases for symmetric graphs.
Limitations
- Constructing a full-rank basis for symmetric graphs remains open; tensor-product models only known to work, scalar methods conjectured infeasible.
- Experiments lack tasks with objective functions of degree l >= 2, so high-degree steerable prediction is not validated empirically.
Experimental setup
Datasets: tetrahedron center prediction (synthetic) 5-body system 100-body system MD17 Water-3D mini completeness test (GWL 2-body, IASR 3-body, k-chain) chirality test (self-designed)
Benchmarks: tetrahedron center prediction 5-body system 100-body system MD17 Water-3D mini completeness test chirality test
Metrics: MSE loss prediction error classification accuracy
Baselines: EGNN FastEGNN HEGNN TFN MACE Equiformer SchNet GVP-GNN
Builds on: EGNN TFN FastEGNN GWL test DeepSet
Closest papers by meaning
- Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks? (NeurIPS 2024 · similarity 0.92)
- A Characterization Theorem for Equivariant Networks with Point-wise Activations (ICLR 2024 · similarity 0.90)
- On the Generalization of Equivariant Graph Neural Networks (ICML 2024 · similarity 0.90)
- On Universality Classes of Equivariant Networks (NeurIPS 2025 · similarity 0.89)
- On Explaining Equivariant Graph Networks via Improved Relevance Propagation (ICML 2025 · similarity 0.89)
- Improving Equivariant Graph Neural Networks on Large Geometric Graphs via Virtual Nodes Learning (ICML 2024 · similarity 0.89)
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