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Continuous Simplicial Neural Networks
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Introduces COSIMO, a continuous simplicial neural network derived from heat-diffusion PDEs on lower and upper Hodge Laplacians, with stability and over-smoothing analysis and experiments on trajectory, mesh regression, node and graph classification.
Problem
Existing simplicial neural networks rely on discrete simplicial filters, which are restrictive, lack control over over-smoothing, and are sensitive to structural perturbations; continuous filtering on simplicial complexes has not been explored.
Approach
Derives a simplicial neural network from a system of PDEs on the lower and upper Hodge Laplacians, whose solutions are matrix exponentials with learnable diffusion times t_d and t_u, giving a dynamic receptive field; the exponentials are approximated via eigendecomposition of the Laplacians.
Key results
- On Shrec-16 mesh regression, COSIMO reaches MSE 0.010 (small) and 0.027 (full), versus 0.020 and 0.063 for the next best SCCNN.
- On node classification, COSIMO scores 0.90 on high-school and 0.69 on senate-bills; on graph classification it scores 0.79 on proteins.
- On the synthetic trajectory dataset COSIMO (0.659) does not beat SCNN (0.677), while on ocean-drifts it reaches 0.550 vs 0.545 for SCCNN.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use COSIMO instead of SCNN | trajectory prediction · synthetic | accuracy: 0.677 ± 0.02 -> 0.659 ± 0.04 | Hurt |
| Use COSIMO instead of SCCNN | trajectory prediction · ocean-drifts | accuracy: 0.545 ± 0.08 -> 0.550 ± 0.06 | Helped |
| Use COSIMO instead of SCCNN | mesh regression · Shrec-16 small | MSE: 0.020 ± 0.003 -> 0.010 ± 0.004 | Helped |
| Use COSIMO instead of SCCNN | mesh regression · Shrec-16 full | MSE: 0.063 ± 0.003 -> 0.027 ± 0.007 | Helped |
| Use COSIMO instead of GSAN | node classification · high-school | accuracy: 0.88 ± 0.05 -> 0.90 ± 0.05 | Helped |
| Use COSIMO instead of GCN | node classification · senate-bills | accuracy: 0.67 ± 0.06 -> 0.69 ± 0.08 | Helped |
| Use COSIMO instead of SaNN | graph classification · proteins | accuracy: 0.77 ± 0.02 -> 0.79 ± 0.01 | Helped |
| Add higher-order convolutions | trajectory prediction · ocean-drifts | accuracy | Helped |
| Add inter-simplicial couplings | trajectory prediction · synthetic | accuracy | No clear effect |
| Set the COSIMO receptive field to t=10^{-2} rather than the discrete SNN setting | COSIMO · over-smoothing analysis | Dirichlet energy convergence rate | Helped |
| Increase the COSIMO receptive field t | COSIMO · over-smoothing analysis | Dirichlet energy convergence rate | Hurt |
Surprising findings
- Inter-simplicial couplings (COSIMO, SCCNN, Bunch) do not outperform SCNN on the synthetic trajectory dataset, attributed to zero input signals on nodes and triangles.
Limitations
- Requires eigendecomposition of Hodge Laplacians, which adds computational cost.
- No clear gain over discrete SCNN on the synthetic trajectory dataset.
Experimental setup
Datasets: synthetic simplicial complex ocean-drifts Shrec-16 high-school senate-bills proteins
Benchmarks: ocean-drifts Shrec-16 high-school senate-bills proteins synthetic simplicial complex
Metrics: accuracy mean squared error Dirichlet energy
Baselines: GCN GraphSAGE GIN GAT SNN SCoNe SCNN Bunch HSN SCACMPS SAN SaNN
Builds on: simplicial convolutional neural network (SCCNN) simplicial filters continuous graph neural networks Hodge Laplacian
Closest papers by meaning
- SNN-PDE: Learning Dynamic PDEs from Data with Simplicial Neural Networks (AAAI 2024 · similarity 0.85)
- CALM-PDE: Continuous and Adaptive Convolutions for Latent Space Modeling of Time-dependent PDEs (NeurIPS 2025 · similarity 0.83)
- PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems (ICLR 2025 · similarity 0.83)
- Breakeven complexity: A new perspective on neural partial differential equation solvers (NeurIPS 2026 · similarity 0.83)
- Simplicial Representation Learning with Neural $k$-Forms (ICLR 2024 · similarity 0.82)
- Smooth Piecewise Cutting for Neural Operator to Handle Discontinuities and Sharp Transitions (NeurIPS 2026 · similarity 0.82)
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