Computational investigations of a multi-class traffic flow model: mean-field and microscopic dynamics

TL;DR

The paper tackles large-scale, multi-class traffic where drivers act non-cooperatively by formulating a Nash mean-field game (NMFG) framework that couples microscopic vehicle dynamics to macroscopic density and velocity fields. It develops an HPC-enabled numerical pipeline using finite-difference discretization and Newton-Krylov solvers with regularization and continuation to solve the resulting forward-backward system, and it recovers microscopic controls from the mean-field solution to assess an epsilon-Nash approximation. Through three two-class scenarios (cars and trucks) and three cost functionals (GLWR, GS, GNS), the study reveals distinct flow regimes and demonstrates macro-micro agreement, supporting the validity of the micro-macro NMFG approach for autonomous-vehicle traffic management. The work provides scalable methods and datasets that inform future multi-class, multi-lane, and second-order traffic models while highlighting practical challenges in preconditioning and non-smooth solution behavior.

Abstract

We address a multi-class traffic model, for which we computationally assess the ability of mean-field games (MFGs) to yield approximate Nash equilibria for traffic flow games of intractable large finite-players. We introduce ad hoc numerical methodologies, with recourse to techniques such as High-Performance Computing (HPC) and regularization of Loose Generalized Minimal Residual (LGMRES) solvers. The developed apparatus allows us to perform simulations at significantly larger space and time discretization scales. For three generic scenarios of cars and trucks, and three cost functionals, we provide numerous numerical results related to the autonomous vehicles (AVs) traffic dynamics, which corroborate for the multi-class case the effectiveness of the approach emphasized in [22]. We additionally provide several original comparisons of macroscopic Nash mean-field speeds with their microscopic versions, allowing us to computationally validate the so-called $ε-$Nash approximation, with a rate slightly better than theoretically expected.

Figures (10)

• Figure 1: Two-class initial density configurations: TC is a fully segregated configuration with cars in front. CT is a fully segregated configuration with trucks in front. TCT is an interlaced configuration with alternating trucks and cars.
• Figure 2: Cost function [GLWR]: evolution of density $\rho_j$, speed $u_j$, and optimal cost $V_j$ for cars and trucks, w.r.t. the position $x$. Each of the first four columns represents a time value, and the last column corresponds to the fundamental diagram. The rows represent the three cases TC, CT, and TCT.
• Figure 3: Cost function [GS]: evolution of density $\rho_j$, speed $u_j$, and optimal cost $V_j$ for cars and trucks, w.r.t. the position $x$. Each of the first four columns represents a time value, and the last column corresponds to the fundamental diagram. The rows represent the three cases TC, CT, and TCT.
• Figure 4: Cost function [GNS]: evolution of density $\rho_j$, speed $u_j$, and optimal cost $V_j$ for cars and trucks, w.r.t. the position $x$. Each of the first four columns represents a time value, and the last column corresponds to the fundamental diagram. The rows represent the three cases TC, CT, and TCT.
• Figure 5: The columns correspond to the TC, CT, and TCT configurations, considering [GS] (left) and [GNS] (right). The rows are for two different values of $n=20,100$. The total number of vehicles $N$ for each column is such that $N=N_1+N_2$, where $N_1=N_2=\alpha n$, with $\alpha=1$ for TC and CT and $\alpha=3$ for TCT. Each sub-figure presents three curves: the $L_{\infty}$ norm $e_v=\left\|\hat{v}-\Bar{v}\right\|_{\infty}$, the cost for NMFE-constructed controls $J_{\hat{v}}=J_{i_{j}}(\hat{v}_{i_{j}},\hat{v}_{-i_{j}},\hat{v}^{-j})$, and the cost for the best response strategies $J_{\bar{v}}=J_{i_{j}}(\Bar{v}_{i_{j}},\hat{v}_{-i_{j}},\hat{v}^{-j})$, all w.r.t. vehicle's index (idx).