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Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework
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Proposes copresheaf topological neural networks (CTNNs), a categorical framework on combinatorial complexes with learnable edge-indexed linear transport maps, unifying GNNs, sheaf NNs and TNNs, and tests small-scale physics, MUTAG and synthetic CC classification experiments.
Problem
Principled design of neural architectures for structured data (images, graphs, meshes, manifolds) with heterogeneous, directional, multiscale structure; existing GNNs oversmooth and assume homophily, transformers assume homogeneous embedding spaces.
Approach
Model each cell with its own feature space and learn directed linear transport maps between cells (copresheaf) over combinatorial complexes, so message passing becomes anisotropic and heterogeneous; shows SNN message passing and diffusion are special cases of this copresheaf message-passing form.
Key results
- On heat equation with transformer, copresheaf attention MSE 9.00e-5 vs classical 2.64e-4 (over 50% reduction).
- On MUTAG, CopresheafSAGE reaches 0.732 accuracy vs GraphSAGE 0.689; CopresheafGCN 0.721 vs GCN 0.674.
- On synthetic CC classification, CT-SharedLoc reaches 0.970 accuracy vs classic transformer 0.940.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace dot-product attention with learned outer-product copresheaf maps | Transformer on Heat · Heat | MSE: 2.64\times10^{-4}\;\pm\,3.50\times10^{-5} -> \mathbf{9.00\times10^{-5}\;\pm\;7.00\times10^{-6}} | Helped |
| Replace dot-product attention with learned outer-product copresheaf maps | Transformer on Advection · Advection | MSE: 3.52\times10^{-4}\,\pm\,7.70\times10^{-5} -> \mathbf{1.20\times10^{-4}\,\pm\,1.20\times10^{-5}} | Helped |
| Replace dot-product attention with learned outer-product copresheaf maps | Conv-transformer on Unsteady Stokes · Unsteady Stokes | MSE: 1.75\times10^{-2}\,\pm\,1.32\times10^{-3} -> \(1.48\times10^{-2}\,\pm\,1.48\times10^{-4}\) | Helped |
| Add learned per-edge copresheaf maps | GCN on MUTAG · MUTAG | accuracy: 0.674 $\pm$ 0.014 -> 0.721 $\pm$ 0.035 | Helped |
| Add learned per-edge copresheaf maps | GraphSAGE on MUTAG · MUTAG | accuracy: 0.689 $\pm$ 0.022 -> 0.732 $\pm$ 0.029 | Helped |
| Add learned per-edge copresheaf maps | GIN on MUTAG · MUTAG | accuracy: 0.700 $\pm$ 0.039 -> 0.724 $\pm$ 0.021 | Helped |
| Add learned per-edge full transport maps | Transformer on CC classification · CC classification | accuracy: 0.940 $\pm$ 0.014 -> 0.955 $\pm$ 0.009 | Helped |
| Use shared transport maps with local modulation rather than full per-edge maps (ablation) | CT-FC on CC classification · CC classification | accuracy: 0.955 $\pm$ 0.009 -> 0.970 $\pm$ 0.010 | Helped |
Limitations
- Per-edge transformations add computational overhead
- Experiments limited to modest-scale settings with small synthetic and benchmark datasets
- Small number of seeds and runs; differences often within reported standard deviations
Experimental setup
Datasets: MUTAG synthetic heat equation data synthetic advection data synthetic unsteady Stokes data synthetic Erdos-Renyi combinatorial complexes
Benchmarks: MUTAG synthetic heat equation synthetic advection synthetic unsteady Stokes synthetic CC classification
Metrics: mean squared error test accuracy
Baselines: GCN GraphSAGE GIN classical transformer dot-product attention classical convolution-transformer U-Net
Builds on: sheaf neural networks topological neural networks combinatorial complexes message passing neural networks transformer attention
Closest papers by meaning
- Topological Blindspots: Understanding and Extending Topological Deep Learning Through the Lens of Expressivity (ICLR 2025 · similarity 0.89)
- TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks (ICML 2025 · similarity 0.88)
- Topological Generalization Bounds for Discrete-Time Stochastic Optimization Algorithms (NeurIPS 2024 · similarity 0.85)
- Topological Neural Networks go Persistent, Equivariant, and Continuous (ICML 2024 · similarity 0.85)
- HOPSE: Scalable Higher-Order Positional and Structural Encoder for Combinatorial Representations (NeurIPS 2026 · similarity 0.85)
- Building Transformation Layers for Riemannian Neural Networks (NeurIPS 2026 · similarity 0.85)
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