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WardropNet: Traffic Flow Predictions via Equilibrium-Augmented Learning
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Proposes WardropNet, a combinatorial-optimization-augmented network that predicts latency parameters with an MLP and feeds them into a Wardrop equilibrium layer trained via a Fenchel-Young loss to predict traffic flows.
Problem
Predicting traffic equilibrium flows quickly and accurately, since computing equilibria with high-fidelity simulators is computationally expensive, and pure ML models fail to capture the combinatorial structure of flows.
Approach
A neural network predicts latency-function parameters, which are passed through a differentiable Wardrop equilibrium layer (regularized by perturbation or Euclidean regularization, or via extended-network min-cost flow for piecewise constant latencies); training uses a Fenchel-Young loss that equals a Bregman divergence and yields usable gradients despite combinatorial layers.
Key results
- CL pipeline outperforms FNN by 60%, 75%, and 7% on HE, LE, and SW stylized scenarios respectively.
- On realistic time-invariant Berlin scenarios the CL pipeline outperforms FNN by 72% and 71%.
- On the time-variant SW-TV scenario the CL pipeline outperforms GNN by 23%.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add a piecewise-constant-latency equilibrium layer to the FNN | FNN · High-entropy (HE) | MAE: 60% | Helped |
| Add a piecewise-constant-latency equilibrium layer to the FNN | FNN · Low-entropy (LE) | MAE: 75% | Helped |
| Add a piecewise-constant-latency equilibrium layer to the FNN | FNN · Square world (SW) | MAE: 7% | Helped |
| Add a piecewise-constant-latency equilibrium layer to the FNN | FNN · Berlin | MAE: 72% | Helped |
| Add a piecewise-constant-latency equilibrium layer to the FNN | FNN · Berlin-AP | MAE: 71% | Helped |
| Add a piecewise-constant-latency equilibrium layer to the GNN | GNN · Square world time-variant (SW-TV) | MAE: 23% | Helped |
| Replace piecewise-constant latencies with polynomial latencies (ablation) | WardropNet · Traffic scenarios | prediction accuracy | Hurt |
| Replace piecewise-constant latencies with Euclidean regularization (ablation) | WardropNet · Traffic scenarios | prediction accuracy | Hurt |
Surprising findings
- The Euclidean-regularized ER pipeline performs worse than pure ML baselines in several settings because its latency representation is too limited.
- The FNN baseline slightly outperforms the GNN in all time-invariant scenarios, despite the GNN using graph structure.
Limitations
- Scaling to larger networks is left as future work.
- Evaluated only on small training sets (9 training instances per scenario).
Experimental setup
Datasets: synthetic traffic equilibria on artificial road network (HE, LE, SW, SW-TV) MATSim simulations for Berlin and Berlin-AP
Benchmarks: HE LE SW SW-TV Berlin Berlin-AP
Metrics: mean absolute error
Baselines: FNN GNN
Builds on: Fenchel-Young loss regularization by perturbation Wardrop equilibrium MATSim
Closest papers by meaning
- Machine-Learned Prediction Equilibrium for Dynamic Traffic Assignment (AAAI 2022 · similarity 0.86)
- Lightweight and Interpretable Transformer via Unrolling of Mixed Graph Algorithms for Traffic Forecast (ICML 2026 · similarity 0.86)
- STDEN: Towards Physics-Guided Neural Networks for Traffic Flow Prediction (AAAI 2022 · similarity 0.85)
- Ising-Traffic: Using Ising Machine Learning to Predict Traffic Congestion under Uncertainty (AAAI 2023 · similarity 0.85)
- Efficient Traffic Prediction Through Spatio-Temporal Distillation (AAAI 2025 · similarity 0.84)
- Closed-Loop Supervised Fine-Tuning of Tokenized Traffic Models (CVPR 2025 · similarity 0.84)
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