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Hyperbolic-PDE GNN: Spectral Graph Neural Networks in the Perspective of A System of Hyperbolic Partial Differential Equations
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Formulates message passing as a system of second-order hyperbolic PDEs on graphs, solves it via a polynomial-approximated Laplacian operator discretized with forward-Euler-style time stepping, and plugs this into classic spectral GNNs to improve node classification and graph signal fitting.
Problem
Standard message passing and spectral GNNs map node features back into a spatial domain unrelated to graph topology, and computing the exact PDE solution needs costly eigendecomposition, limiting interpretability and flexibility.
Approach
Model message passing as a second-order hyperbolic PDE on the graph whose solution space is spanned by Laplacian eigenvectors; approximate the Laplacian operator with an orthogonal polynomial in L and discretize time, so any spectral GNN's polynomial filter can serve as the operator inside the hyperbolic update.
Key results
- Hyperbolic-Cheb reaches 89.38% on Cora and 90.50% on PubMed, versus 86.67% and 87.95% for ChebNet, with gains of about 2-3 points on homophilic citation graphs.
- Hyperbolic-Cheb improves Texas accuracy from 86.22% to 93.93%, a gain of 7.71 points.
- On the image signal filtering task, Hyperbolic-GCN achieves lower squared error than GCN on all seven filters, e.g. a 97.38% reduction on high-pass.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| apply hyperbolic-PDE message passing to GPR-GNN | GPR-GNN · Cora | accuracy: +2.25 | Helped |
| apply hyperbolic-PDE message passing to GPR-GNN | GPR-GNN · CiteSeer | accuracy: +1.76 | Helped |
| apply hyperbolic-PDE message passing to GPR-GNN | GPR-GNN · PubMed | accuracy: +2.90 | Helped |
| apply hyperbolic-PDE message passing to GPR-GNN | GPR-GNN · Computers | accuracy: +1.36 | Helped |
| apply hyperbolic-PDE message passing to GPR-GNN | GPR-GNN · Photo | accuracy: +0.64 | Helped |
| apply hyperbolic-PDE message passing to GPR-GNN | GPR-GNN · CS | accuracy: +0.11 | Helped |
| apply hyperbolic-PDE message passing to GPR-GNN | GPR-GNN · Actor | accuracy: +0.76 | Helped |
| apply hyperbolic-PDE message passing to GPR-GNN | GPR-GNN · Texas | accuracy: +0.33 | Helped |
| apply hyperbolic-PDE message passing to GPR-GNN | GPR-GNN · Cornell | accuracy: +1.40 | Helped |
| apply hyperbolic-PDE message passing to GPR-GNN | GPR-GNN · DeezerEurope | accuracy: ↓ 0.77 | Hurt |
| apply hyperbolic-PDE message passing to ChebNet | ChebNet · Low-pass filter | R² score: 0.8024(.9973) -> 0.0077(1.000) | Helped |
| apply hyperbolic-PDE message passing to ChebNet | ChebNet · High-pass filter | R² score: 0.7877(.9903) -> 0.0088(.9999) | Helped |
Surprising findings
- Hyperbolic-PDE variants reduce accuracy on some datasets, e.g. Hyperbolic-SGC drops 2.02 points on CiteSeer and Hyperbolic-Bern drops 1.83 points on Actor.
Limitations
- Some base models (e.g. Hyperbolic-SGC on CiteSeer, Hyperbolic-Bern on Actor) show accuracy decreases, so gains are not uniform.
- Theoretical claims about eigenvector-spanned solution space rest on a simplified derivation with a vanilla Laplacian; efficiency gains are not empirically quantified.
Experimental setup
Datasets: Cora CiteSeer PubMed Computers Photo CS Actor Texas Cornell DeezerEurope 50 Matlab images (100x100 grid graphs)
Benchmarks: Cora CiteSeer PubMed Computers Photo CS Actor Texas Cornell DeezerEurope
Metrics: accuracy R^2 score squared error
Baselines: GCN SGC APPNP ARMA GPR-GNN ChebNet BernNet EvenNet JacobiConv ChebNetII OptBasisGNN PyGNN
Builds on: GCN SGC APPNP GPR-GNN ChebNet BernNet JacobiConv ChebNetII GraphCON GRAND
Closest papers by meaning
- Graph Neural PDE Solvers with Conservation and Similarity-Equivariance (ICML 2024 · similarity 0.90)
- PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems (ICLR 2025 · similarity 0.88)
- Position: Spectral GNNs Rely Less on Graph Fourier Basis than Conceived (ICML 2025 · similarity 0.88)
- Enhancing Spectral GNNs: From Topology and Perturbation Perspectives (ICML 2025 · similarity 0.88)
- WaveNet: Tackling Non-stationary Graph Signals via Graph Spectral Wavelets (AAAI 2024 · similarity 0.87)
- Permutation Equivariant Neural Controlled Differential Equations for Dynamic Graph Representation Learning (NeurIPS 2025 · similarity 0.87)
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