Advection Dynamics in Traffic Networks: Modeling, Analysis, and Optimization

Traffic congestion is a critical challenge in urban environments, negatively affecting quality of life and economic productivity. Mathematical models have sought to provide foundational insights, with microscopic models capturing individual behaviors and macroscopic models offering computational efficiency. Microscopic models, however, are computationally intensive, while macroscopic models lack the detail required for devising traffic solutions. This work introduces a novel, intermediate-resolution framework based on networks of dynamic systems governed by ordinary differential equations with discontinuous right-hand sides. Our approach employs graph-based advection dynamics, where nodes represent discrete road segments with capacity constraints. Through analytical derivations, we gain insights into the equilibrium states that emerge as roads experience congestion. We further exploit the mathematical tractability of our model to formulate well-posed optimization problems designed to mitigate congestion. The efficacy of our framework is validated using both representative numerical examples and a real-world dataset from Manhattan's Upper West Side in New York City, USA. Notably, our findings suggest that sparse, time-dependent interventions on critical road segments can be used to effectively alleviate congestion throughout the day.