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Simplicial Representation Learning with Neural $k$-Forms
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Proposes neural k-forms, learnable differential k-forms parameterized by MLPs that integrate over embedded simplices to produce message-passing-free representations (integration matrices) of geometric graphs and simplicial complexes.
Problem
Geometric deep learning on graphs and simplicial complexes relies on message passing, which suffers from over-smoothing, over-squashing, and needs graph rewiring; the paper targets learning representations of embedded simplicial complexes using node coordinates without message passing.
Approach
Represent each embedded k-simplex by its integrals against learnable neural k-forms (MLP-parameterized scaling functions in R^n), yielding an integration matrix whose columns share a common interpretation across complexes; proves a universal approximation result for neural k-forms.
Key results
- On MoleculeNet, NkF achieves AUROC 83.50 on BACE, 86.41 on BBBP, and 76.70 on HIV with about 9K parameters, outperforming GAT, GCN, and GIN.
- On small TU graph datasets, NkF reaches 93.42 on Letter-low and 67.69 on Letter-med, best among non-equivariant models, while being competitive on BZR, COX2, and DHFR with ~4K parameters.
- Learned 1-forms on synthetic path classification and 2-forms on surface classification yield clearly separated class representations.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use neural k-forms instead of GAT | Graph classification · BZR | accuracy: 81.48\pm2.90 -> 78.77\pm0.55 | Hurt |
| Use neural k-forms instead of GCN | Graph classification · COX2 | accuracy: 79.88\pm1.65 -> 80.30\pm2.42 | Helped |
| Use neural k-forms instead of GAT | Graph classification · DHFR | accuracy: 73.02\pm2.54 -> 64.42\pm2.09 | Hurt |
| Use neural k-forms instead of EGNN | Graph classification · Letter-low | accuracy: 93.20\pm0.68 -> 93.42\pm1.94 | No clear effect |
| Use neural k-forms instead of GIN | Graph classification · Letter-med | accuracy: 67.07\pm2.47 -> 67.69\pm1.28 | No clear effect |
| Use neural k-forms instead of EGNN | Graph classification · Letter-high | accuracy: 68.98\pm 1.80 -> 62.93\pm4.13 | Hurt |
| Use neural k-forms instead of GAT | Graph classification · BACE | AUROC: 69.52\pm17.52 -> 83.50\pm0.55 | Helped |
| Use neural k-forms instead of GAT | Graph classification · BBBP | AUROC: 76.51\pm 3.36 -> 86.41\pm 3.64 | Helped |
| Use neural k-forms instead of GCN | Graph classification · HIV | AUROC: 68.70\pm 1.67 -> 76.70\pm 2.17 | Helped |
Surprising findings
- On MoleculeNet, the GNN baselines show large variance (e.g., GIN BACE 42.91±18.56), and the paper reports that smaller GNNs suffered significant drops in performance.
Limitations
- Requires node coordinates, so only applies to embedded complexes and cannot handle coordinate-equivariant node features.
- Number of monomial k-forms binom(n,k) and numerical integration cost grow for high-dimensional embeddings and higher k.
- Only benchmarked on graph classification; regression and higher-dimensional simplicial benchmarks untested.
Experimental setup
Datasets: BZR COX2 DHFR Letter-low Letter-med Letter-high BACE BBBP HIV synthetic path data synthetic surface data
Benchmarks: BZR COX2 DHFR Letter-low Letter-med Letter-high BACE BBBP HIV
Metrics: accuracy AUROC
Baselines: EGNN GAT GCN GIN
Builds on: Geometric deep learning Universal Approximation Theorem simplicial neural networks
Closest papers by meaning
- Geometry Awakening: Cross-Geometry Learning Exhibits Superiority over Individual Structures (NeurIPS 2024 · similarity 0.87)
- Cover learning for large-scale topology representation (ICML 2025 · similarity 0.86)
- Unsupervised Parameter-free Simplicial Representation Learning with Scattering Transforms (ICML 2024 · similarity 0.85)
- On the Completeness of Invariant Geometric Deep Learning Models (ICLR 2025 · similarity 0.85)
- Topological Blindspots: Understanding and Extending Topological Deep Learning Through the Lens of Expressivity (ICLR 2025 · similarity 0.84)
- Distilling Graph Geometry: Knowledge Gap from GNNs to MLPs (NeurIPS 2026 · similarity 0.84)
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