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Topological Neural Networks go Persistent, Equivariant, and Continuous
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Proposes TopNets, a unified framework combining topological neural networks with persistent homology, extended to E(n)-equivariant and continuous (neural ODE) variants, evaluated on graph classification, QM9, CDR-H3 antibody design, and 3BPA MD17.
Problem
Integrating persistent homology descriptors into topological neural networks (TNNs) to boost expressivity, and extending such models to geometric complexes with E(n) symmetry and continuous-depth dynamics.
Approach
A general TopNet layer (message passing on simplicial complexes, then persistence diagram vectorization with topological aggregation and readouts) that subsumes TOGL, PersLay, and RePHINE; equivariant variants use invariant geometric simplex-color filtrations, and continuous variants are Neural ODEs with derived discretization error bounds.
Key results
- On graph classification, continuous TopNets (GCN with RePHINE) improve over discrete counterparts on all datasets, e.g., IMDB-B 69.40 to 76.00 and NCI1 80.44 to 82.15.
- On CDR-H3 antibody design, TopNets (RePHINE diagrams) reach 44.80% AAR versus 39.8% for AbODE, with RMSD 1.75 vs 1.73.
- On 3BPA MD17 at 300K, TopNets achieve energy RMSE 2.5 meV versus 3.0 for MACE and 3.3 for NequIP; force is comparable to MACE (8.9 vs 8.8).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use continuous rather than discrete TopNets (ablation) | TopNets on graph classification · Graph classification datasets | predictive performance | Helped |
| Use continuous rather than discrete TopNets (ablation) | GCN TopNets with vertex-color diagrams · Graph classification runtime | GPU time: 62.9 -> 106.8 | Hurt |
| Use continuous rather than discrete TopNets (ablation) | GCN TopNets with RePHINE diagrams · Graph classification runtime | GPU time: 62.7 -> 112.7 | Hurt |
| Use continuous rather than discrete TopNets (ablation) | GIN TopNets with vertex-color diagrams · Graph classification runtime | GPU time: 60.7 -> 105.7 | Hurt |
| Use continuous rather than discrete TopNets (ablation) | GIN TopNets with RePHINE diagrams · Graph classification runtime | GPU time: 65.9 -> 120.7 | Hurt |
| Use RePHINE diagrams rather than vertex-color diagrams (ablation) | TopNets for CDR-H3 design · CDR-H3 design | AAR (%): 43.00 -> 44.80 | Helped |
| Use RePHINE diagrams rather than vertex-color diagrams (ablation) | TopNets for CDR-H3 design · CDR-H3 design | RMSD: 1.73 -> 1.75 | Mixed |
| Add persistent homology to the model | TopNets for antibody design · CDR-H3 design | AAR (%): 39.8 -> 44.80 | Helped |
| Use TopNets rather than MACE | 3BPA molecular dynamics prediction · 300K | RMSE energy (meV): 3.0±0.2 -> 2.5±0.2 | Helped |
| Use TopNets rather than MACE | 3BPA molecular dynamics prediction · 300K | RMSE force (meV/Å): 8.8±0.3 -> 8.9 ±0.2 | Mixed |
| Use TopNets rather than MACE | 3BPA molecular dynamics prediction · 600K | RMSE energy (meV): 9.7±0.5 -> 9.5±0.7 | Helped |
| Use TopNets rather than MACE | 3BPA molecular dynamics prediction · 600K | RMSE force (meV/Å): 21.8±0.6 -> 21.8 ±0.7 | No clear effect |
Surprising findings
- Persistence diagrams from geometric 0-simplex-color filtrations are not more expressive than vertex-color ones, since the only E(n)-invariant function of a single point is constant.
- Continuous ODE-based variants require roughly 1.7-2x the per-epoch GPU time of discrete counterparts.
Limitations
- Computing persistent homology embeddings and higher-order message passing is computationally expensive.
- Scope restricted to simplicial complexes; combinatorial complexes not explored.
- Discretization error bounds are not guaranteed to vanish, since it is unclear whether R_1(h)N tends to zero.
Experimental setup
Datasets: NCI109 IMDB-BINARY NCI1 ogbg-molhiv PROTEINS ENZYMES DD QM9 SAbDab 3BPA MD17
Benchmarks: NCI109 IMDB-B NCI1 MOLHIV PROTEINS QM9 CDR-H3 SAbDab 3BPA MD17
Metrics: accuracy AUROC MAE AAR RMSD RMSE
Baselines: GCN GIN MPSN TOGL RePHINE EGNN IMPSN MACE NequIP AbODE MEAN RefineGNN
Builds on: TOGL RePHINE PersLay EMPSN GRAND
Closest papers by meaning
- Topological Blindspots: Understanding and Extending Topological Deep Learning Through the Lens of Expressivity (ICLR 2025 · similarity 0.89)
- TopoGCL: Topological Graph Contrastive Learning (AAAI 2024 · similarity 0.88)
- TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration (ICML 2025 · similarity 0.88)
- TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks (ICML 2025 · similarity 0.88)
- TopER: Topological Embeddings in Graph Representation Learning (NeurIPS 2025 · similarity 0.87)
- Positional Encoding meets Persistent Homology on Graphs (ICML 2025 · similarity 0.87)
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