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Topology-aware Neural Flux Prediction Guided by Physics
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Proposes PhyNFP, which replaces adjacency with learnable upwind discretized difference matrices and embeds Saint-Venant and Aw-Rascle PDE updates into GNN message passing to improve directional sensitivity in river and traffic flux prediction.
Problem
GNNs act as low-pass filters on directed graphs and thus fail to preserve high-frequency nodal signals, making them insensitive to edge directions in flow dynamics such as river flux and traffic flow prediction.
Approach
Replace the adjacency operator with upwind discretized difference matrices (scaled by learnable distance and elevation mappings) and build message passing from physical PDE updates (Saint-Venant momentum for rivers, Aw-Rascle mass conservation for traffic), so that GNN layers retain high-frequency, direction-dependent information.
Key results
- On the river network with horizon n=6, PhyNFP achieves forward MSE 0.0801 versus an average of 0.1170 for baseline GNNs, a reported 31.6% reduction.
- On the traffic network, PhyNFP achieves forward MSE 0.0696 versus a baseline average of 0.0732 (4.9% reduction), with DS 0.0028 vs. baseline 0.0006.
- Direction sensitivity DS on the river network is 0.0105 for PhyNFP versus an average of about 0.0004 for baselines.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add PDE constraints to the difference-matrix model (ablation) | PhyNFP · River Network, Forward | MSE: 0.0898 -> 0.0801 | Helped |
| add PDE constraints to the difference-matrix model (ablation) | PhyNFP · Traffic Network, Forward | MSE: 0.0721 -> 0.0696 | Helped |
| add PDE constraints to the difference-matrix model (ablation) | PhyNFP · River Network, Forward | DS: +0.0063 -> +0.0105 | Helped |
| add PDE constraints to the difference-matrix model (ablation) | PhyNFP · Traffic Network, Forward | DS: +0.0017 -> +0.0028 | Helped |
| use PhyNFP instead of the strongest Forward-MSE baseline | River Network · River Network, Forward (n=6) | MSE: 0.1126 -> 0.0801 | Helped |
| use PhyNFP instead of the strongest Forward-MSE baseline | Traffic Network · Traffic Network, Forward (n=6) | MSE: 0.0700 -> 0.0696 | Helped |
| use PhyNFP instead of graph learners on average | River Network · River Network, Forward | MSE: 31.6% reduction | Helped |
| use PhyNFP instead of graph learners on average | Traffic Network · Traffic Network, Forward | MSE: 4.9% reduction | Helped |
| increase prediction horizon from n=3 to n=9 | PhyNFP · River Network, Forward | MSE: 0.0514 -> 0.1087 | Hurt |
| increase prediction horizon from n=3 to n=9 | GAT · River Network, Forward | MSE: 0.0617 -> 0.1433 | Hurt |
| increase prediction horizon from n=3 to n=9 | GCN · River Network, Forward | MSE: 0.0632 -> 0.1481 | Hurt |
Surprising findings
- Baseline GNNs (GCN, GAT, GraphSAGE, MPNN) show negative or near-zero direction sensitivity, i.e., they perform as well or better on reversed topology than on forward topology.
- Traffic network shows much weaker direction sensitivity than the river network, attributed to cycles in the graph and the mass-conservation physics.
Limitations
- Physical equations are simplified (e.g., water depth and friction neglected in the Saint-Venant case) and boundary/initial conditions are not incorporated.
- Evaluated on only two preprocessed datasets with a single prediction horizon in the main table.
- Direction sensitivity is weaker on the traffic network, where cyclic structure counteracts directionality.
Experimental setup
Datasets: LamaH-CE2 (Danube river network, preprocessed) PEMS-04 (traffic flow, preprocessed)
Benchmarks: LamaH-CE2 river network (preprocessed) PEMS-04 traffic network (preprocessed)
Metrics: MSE direction sensitivity (DS) relative direction sensitivity (RDS)
Baselines: GCN GAT MPNN GraphSAGE GWN MP-PDE Solver GNO PhyNFP_DM
Builds on: Message-Passing PDE Solver Graph Wavelet Network PDE-GCN GraphCON
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 0.86)
- Graph Neural Networks for Edge Signals: Orientation Equivariance and Invariance (ICLR 2025 · similarity 0.86)
- Weisfeiler and Lehman Go Paths: Learning Topological Features via Path Complexes (AAAI 2024 · similarity 0.85)
- The Merit of River Network Topology for Neural Flood Forecasting (ICML 2024 · similarity 0.85)
- Feature Transportation Improves Graph Neural Networks (AAAI 2024 · similarity 0.85)
- Topology-Aware Conformal Prediction for Stream Networks (NeurIPS 2025 · similarity 0.85)
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