# WardropNet: Traffic Flow Predictions via Equilibrium-Augmented Learning

arXiv: https://arxiv.org/abs/2410.06656 · HTML: https://mlsanity.com/paper/2410.06656/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 48 references that are in the map.
- First posted on arXiv 9 Oct 2024.

## Builds on (most cited first, 17 of 17)

- [Spatio-Temporal Graph Convolutional Networks: A Deep Learning Framework for Traffic Forecasting](https://mlsanity.com/paper/1709.04875/) (2017)
- [Graph WaveNet for Deep Spatial-Temporal Graph Modeling](https://mlsanity.com/paper/1906.00121/) (2019)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Adaptive Graph Convolutional Recurrent Network for Traffic Forecasting](https://mlsanity.com/paper/2007.02842/) (2020)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [Spatio-Temporal Graph Neural Networks for Predictive Learning in Urban Computing: A Survey](https://mlsanity.com/paper/2303.14483/) (2023)
- [Learning with Fenchel-Young Losses](https://mlsanity.com/paper/1901.02324/) (2019)
- [Implicit Graph Neural Networks](https://mlsanity.com/paper/2009.06211/) (2020)
- [Decision Trees for Decision-Making under the Predict-then-Optimize Framework](https://mlsanity.com/paper/2003.00360/) (2020)
- [Learning with Combinatorial Optimization Layers: a Probabilistic Approach](https://mlsanity.com/paper/2207.13513/) (2022)
- [End-to-End Learning and Intervention in Games](https://mlsanity.com/paper/2010.13834/) (2020)
- [Learning-based Online Optimization for Autonomous Mobility-on-Demand Fleet Control](https://mlsanity.com/paper/2302.03963/) (2023)
- [Turbocharging Solution Concepts: Solving NEs, CEs and CCEs with Neural Equilibrium Solvers](https://mlsanity.com/paper/2210.09257/) (2022)
- [Combinatorial Optimization enriched Machine Learning to solve the Dynamic Vehicle Routing Problem with Time Windows](https://mlsanity.com/paper/2304.00789/) (2023)
- [Differentiable Equilibrium Computation with Decision Diagrams for Stackelberg Models of Combinatorial Congestion Games](https://mlsanity.com/paper/2110.01773/) (2021)

## Built on by (most cited first, 3 of 3)

- [Learning traffic flows: Graph Neural Networks for Metamodelling Traffic Assignment](https://mlsanity.com/paper/2505.11230/) (2025)
- [Combinatorial Optimization Augmented Machine Learning](https://mlsanity.com/paper/2601.10583/) (2026)
- [Learning from user's behaviour of some well-known congested traffic networks](https://mlsanity.com/paper/2508.14804/) (2025)

## Abstract (arXiv)

When optimizing transportation systems, anticipating traffic flows is a central element. Yet, computing such traffic equilibria remains computationally expensive. Against this background, we introduce a novel combinatorial optimization augmented neural network architecture that allows for fast and accurate traffic flow predictions. We propose WardropNet, a neural network that combines classical layers with a subsequent equilibrium layer: the first ones inform the latter by predicting the parameterization of the equilibrium problem's latency functions. Using supervised learning we minimize the difference between the actual traffic flow and the predicted output. We show how to leverage a Bregman divergence fitting the geometry of the equilibria, which allows for end-to-end learning. WardropNet outperforms pure learning-based approaches in predicting traffic equilibria for realistic and stylized traffic scenarios. On realistic scenarios, WardropNet improves on average for time-invariant predictions by up to 72% and for time-variant predictions by up to 23% over pure learning-based approaches.