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E(n) Equivariant Topological Neural Networks
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Introduces E(n)-equivariant message passing networks over combinatorial complexes (ETNNs), applied to QM9 molecular property prediction and a new multi-resolution geospatial air-pollution downscaling benchmark.
Problem
Topological deep learning lacks a principled way to incorporate geometric features (positions, velocities) with E(n) equivariance over general combinatorial topological spaces beyond graphs and simplicial complexes.
Approach
Scalarization-based E(n)-equivariant message passing over combinatorial complexes: invariant geometric features (pairwise distances, centroid distances, Hausdorff distance, convex hull volume) feed learnable messages per neighborhood, with equivariant position updates; EGNN and EMPSN are recovered as special cases.
Key results
- On QM9, ETNN-* achieves MAE of 0.062 for alpha vs 0.071 for EGNN (-13%) and 0.022 for mu (-26%), state of the art on mu, matching or surpassing EMPSN on most properties.
- ETNN uses about 2 orders of magnitude fewer cells per molecule (39.8 vs 555.25 for EMPSN's Vietoris-Rips complex), with about half the memory and almost half the runtime per epoch of EMPSN.
- On air pollution downscaling, ETNN achieves average R^2 of 9.34% vs 1.43% for EGNN, 2.44% for GNN and 2.35% for MLP (30 seeds).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use ETNN-* instead of EGNN | QM9 molecular property prediction · QM9 μ | MAE: .029 -> .022 | Helped |
| use ETNN-* instead of EGNN | QM9 molecular property prediction · QM9 α | MAE: .071 -> .062 | Helped |
| use ETNN-* instead of Equiformer | QM9 molecular property prediction · QM9 Δε | MAE: 30 -> 45 | Hurt |
| use ETNN-* instead of EMPSN* | QM9 molecular property prediction · QM9 α | MAE: .061 -> .062 | Hurt |
| add the virtual cell (ablation) | ETNN · QM9 α | MAE: .161 -> .062 | Helped |
| add the virtual cell (ablation) | ETNN · QM9 μ | MAE: .306 -> .022 | Helped |
| use ETNN instead of GNN | Air pollution downscaling · Air pollution downscaling | AEV (R²): 2.44% -> 9.34% | Helped |
| use ETNN instead of GNN | Air pollution downscaling · Air pollution downscaling | MSE: 0.987 -> 0.935 | Helped |
| add the virtual cell (ablation) | ETNN · Air pollution downscaling | DEV (R²): -1.80% | Helped |
| add the position update (ablation) | ETNN · Air pollution downscaling | DEV (R²): -1.37% | Helped |
| add geometric features (ablation) | ETNN · Air pollution downscaling | DEV (R²): -1.08% | Helped |
Surprising findings
- Without the virtual cell, ETNN performance drops sharply on QM9 (e.g., mu MAE 0.306), highlighting the importance of the virtual cell.
- On the air pollution task, the invariant ETNN without position updates slightly outperformed the variant that used features but not position updates.
Limitations
- Cannot directly handle time-varying scenarios.
- Scalarization-based invariants are hand-designed; going beyond scalarization is left open.
- Scalability bottlenecks from message passing and convex hull volume computation on large datasets.
Experimental setup
Datasets: QM9 mobile air sensor PM2.5 measurements (Bronx, NY, Q4 2021) census tract / road / point geospatial features
Benchmarks: QM9 air pollution downscaling
Metrics: mean absolute error R^2 (explained variance) mean squared error runtime memory usage
Baselines: EGNN EMPSN Equiformer DimeNet++ TopNet GNN MLP Linear VN-EGNN
Builds on: EGNN EMPSN CCMPN TopNets
Closest papers by meaning
- TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks (ICML 2025 · similarity 0.88)
- Universally Invariant Learning in Equivariant GNNs (NeurIPS 2025 · similarity 0.88)
- Topological Blindspots: Understanding and Extending Topological Deep Learning Through the Lens of Expressivity (ICLR 2025 · similarity 0.88)
- On Explaining Equivariant Graph Networks via Improved Relevance Propagation (ICML 2025 · similarity 0.87)
- Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks? (NeurIPS 2024 · similarity 0.87)
- Topological Neural Networks go Persistent, Equivariant, and Continuous (ICML 2024 · similarity 0.87)
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