A nonlocal degenerate macroscopic model of traffic dynamics with saturated diffusion: modeling and calibration theory
Dawson Do, Hossein Nick Zinat Matin, Masuma Mollika Miti, Maria Laura Delle Monache
TL;DR
This work tackles limitations of classical Lighthill–Whitham–Richards traffic models by introducing a first-order nonlocal PDE with saturated diffusion, where the flux depends on a nonlocally perceived density $\hat{\rho}$ built from the local density $\rho$, diffusion $D(\rho)=\rho(1-\rho)$, and a bounded saturating function $\Psi$. The model’s velocity uses a nonlocal kernel $\mathcal{K}_\gamma$, enabling look-ahead behavior, while diffusion degenerates at extreme densities to preserve physical advection. A calibration framework is developed to align the PDE’s deterministic semigroup with the empirical probability transition kernel observed in data, offering two pathways: fundamental-diagram-based calibration and solution-based calibration. Across PeMS and drone datasets, the nonlocal, saturating-diffusion model demonstrates improved ability to reproduce evolving density distributions and congestion dynamics, with nonzero diffusion and short-range look-ahead frequently yielding the best fits. This approach provides a principled link between macroscopic PDE dynamics and stochastic traffic-state evolution, with practical implications for calibration, prediction, and understanding driver behavior.
Abstract
In this work, we introduce a novel first-order nonlocal partial differential equation with saturated diffusion to describe the macroscopic behavior of traffic dynamics. We show how the proposed model is better in comparison with existing models in explaining the underlying driver behavior in real traffic data. In doing so, we introduce a methodology for adjusting the parameters of the proposed PDE with respect to the distribution of real datasets. In particular, we conceptually and analytically elaborate on how such calibration connects the solution of the PDE to the probability transition kernel proposed by the datasets. The performance of the model is thoroughly investigated with respect to several metrics. More precisely, we study the capability of the model in capturing the probability distribution realized by the datasets in the form of the fundamental diagram. We show that the model is capable of approximating the dynamics of the evolution of the probability distribution. To this end, we evaluate the performance of the model with regard to the congestion formation and dissipation scenarios from various datasets.