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Continuous Product Graph Neural Networks
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Introduces CITRUS, a continuous graph neural network that solves tensorial PDEs on Cartesian product graphs via separable heat kernels, with stability and over-smoothing analyses, evaluated on traffic and weather spatiotemporal forecasting.
Problem
Existing product-graph GNNs use discrete graph filters that need costly grid searches, have limited receptive fields and are restricted to two factor graphs, while continuous GNNs handle only a single graph. The paper targets joint learning over multidomain data on multiple interacting graphs.
Approach
Defines tensorial PDEs on graphs (TPDEGs) whose solution is a separable heat kernel over the factor graphs of a Cartesian product; CITRUS implements this with learnable continuous receptive fields and per-factor eigendecompositions, giving a general number of factor graphs.
Key results
- On MetrLA at H=12, CITRUS reports MAE 3.44 and RMSE 6.85, the best among the compared methods in that table.
- On PemsBay at H=6, CITRUS reports MAE 1.48 versus 1.54 for the next-best GRUGCN.
- On NOAA rNMSE at H=1, CITRUS reports 0.04 versus 0.15 for the next-best GRUGCN.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use joint product-graph learning instead of sequential TTS (ablation) | CITRUS · MetrLA H=12 | MAE: 3.67 -> 3.44 | Helped |
| use joint product-graph learning instead of sequential STT (ablation) | CITRUS · MetrLA H=12 | MAE: 3.65 -> 3.44 | Helped |
| replace discrete GNN with a continuous GNN in the TTS pipeline (ablation) | TTS · MetrLA H=12 | MAE: 3.67 -> 3.61 | Helped |
| replace discrete GNN with a continuous GNN in the STT pipeline (ablation) | STT · MetrLA H=12 | MAE: 3.65 -> 3.63 | Helped |
| increase selected eigenpair count from k=2 to k=200 (ablation) | CITRUS · MetrLA | MAE: 2.80 -> 2.70 | Helped |
| increase selected eigenpair count from k=2 to k=200 (ablation) | CITRUS · MetrLA | training time (seconds): 2.52 -> 7.78 | Hurt |
| use k=50 selected eigenpairs rather than k=200 (ablation) | CITRUS · MetrLA | MAE: 2.71 -> 2.70 | No clear effect |
| use k=50 selected eigenpairs rather than k=200 (ablation) | CITRUS · MetrLA | training time (seconds): 7.78 -> 3.86 | Helped |
| use CITRUS rather than GRUGCN | traffic forecasting · MetrLA H=12 | MAE: 3.62 -> 3.44 | Helped |
| use CITRUS rather than GMAN | traffic forecasting · PemsBay H=12 | MAE: 1.86 -> 1.78 | Helped |
| use CITRUS rather than SGP | weather forecasting · Molene H=1–5 | rNMSE: 0.24, 0.37, 0.49, 0.59, 0.69 -> 0.23, 0.35, 0.47, 0.58, 0.67 | Helped |
| use CITRUS rather than GRUGCN | weather forecasting · NOAA H=1–5 | rNMSE: 0.15, 0.18, 0.24, 0.30, 0.37 -> 0.04, 0.12, 0.20, 0.29, 0.36 | Helped |
Surprising findings
- Over-smoothing bound is loose when the spectral condition is violated (right plot), which the authors acknowledge.
- On MetrLA, CITRUS's MAPE at H=3 (6.74%) is worse than GRUGCN's 6.61% and the MAE is tied with Graph WaveNet and GTCNN, so it does not win every metric.
Limitations
- Stability and over-smoothing bounds are loose in some regimes, and tighter bounds are left to future work.
- Experiments on more than two factor graphs are limited to small synthetic settings.
- Performance is not uniformly best; on some metrics such as MetrLA MAPE at H=3, baselines are better.
Experimental setup
Datasets: MetrLA PemsBay Molene NOAA
Benchmarks: MetrLA PemsBay Molene NOAA
Metrics: MAE MAPE RMSE rNMSE
Baselines: ARIMA G-VARMA GP-VAR FC-LSTM Graph WaveNet GMAN STGCN GGRNN GRUGCN SGP GTCNN
Builds on: graph-time convolutional neural network continuous graph neural networks graph neural diffusion Dirichlet energy over-smoothing analysis
Closest papers by meaning
- Hyperbolic-PDE GNN: Spectral Graph Neural Networks in the Perspective of A System of Hyperbolic Partial Differential Equations (ICML 2025 · similarity 0.87)
- Designing Specialized Two-Dimensional Graph Spectral Filters for Spatial-Temporal Graph Modeling (AAAI 2025 · similarity 0.87)
- Spatio-Spectral Graph Neural Networks (NeurIPS 2024 · similarity 0.85)
- Graph Neural PDE Solvers with Conservation and Similarity-Equivariance (ICML 2024 · similarity 0.85)
- PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems (ICLR 2025 · similarity 0.85)
- Uncertainty Estimation on Graphs with Structure Informed Stochastic Partial Differential Equations (NeurIPS 2025 · similarity 0.85)
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