[Paper Review] Non-separable Mean Field Games for Pedestrian Flow: Generalized Hughes Model

Mohamed Ghattassi, Nader Masmoudi|arXiv (Cornell University)|Oct 7, 2023

Traffic control and management4 citations

TL;DR

This paper introduces a generalized Hughes model for pedestrian flow using non-separable mean field games, where individuals avoid or navigate high-density regions based on local density knowledge. It establishes the existence of weak solutions and analyzes the vanishing viscosity limit, with numerical experiments showing that the parameter β governs behavior—ranging from avoidance (β=2) to attraction (β=0) in crowd dynamics.

ABSTRACT

In this paper, we present a new generalized Hughes model designed to intelligently depict pedestrian congestion dynamics, allowing pedestrian groups to either navigate through or circumvent high-density regions. First, we describe the microscopic settings of the model. The corresponding optimization problems are deterministic and can be formulated by a closed-loop model predictive control strategy. This microscopic setup leads in the mean-field limit to the Generalized Hughes model which is a class of non-separable mean field games system, i.e., Fokker-Planck equation and viscous Hamilton-Jacobi Bellman equation are coupled in a forward-backward structure. We give an overview on the mean field games in connection to our intelligent fluid model. Therefore, we show the existence of weak solutions to the Generalized Hughes model and analyze the vanishing viscosity limit of weak solutions. Finally, we illustrate the generalized Hughes model with various numerical experiments.

Motivation & Objective

• To develop a generalized Hughes model that captures intelligent pedestrian behavior in response to local density, allowing for both avoidance and navigation through high-density regions.
• To establish a rigorous mathematical framework linking microscopic control strategies to a macroscopic non-separable mean field game system.
• To prove the existence of weak solutions to the generalized Hughes model and analyze the vanishing viscosity limit.
• To numerically validate the model’s ability to reproduce realistic crowd dynamics, including dispersion, diffusion, and concentration phenomena.
• To investigate the role of the parameter β in regulating pedestrian behavior—ranging from avoidance to attraction to high-density regions.

Proposed method

• Formulate a microscopic model based on closed-loop model predictive control, where each pedestrian optimizes their path under partial knowledge of local density.
• Derive the mean-field limit to obtain a macroscopic system coupling a viscous Hamilton-Jacobi-Bellman equation (forward) with a Fokker-Planck equation (backward), forming a non-separable mean field game.
• Use a regularized approximation scheme to handle low regularity of the potential φ, enabling existence and uniqueness proofs for weak solutions.
• Apply a vanishing viscosity approach to analyze the limit of weak solutions as the viscosity parameter tends to zero.
• Implement numerical simulations using FreeFem++ with triangular finite element meshes (h = 1/40) and explicit time stepping (dt = 5×10⁻²), solving the system over T = 50.
• Vary the parameter β in the speed function f(ρ) = f(ρ)^(1−β) to simulate different behavioral regimes: avoidance (β=2), indifference (β=1), and attraction (β=0).

Figure 1: Schematic illustration of density contour levels (a) and velocity field (b) at the initial instant $\displaystyle t=0$ ( $\displaystyle\beta=2$ ).

Experimental results

Research questions

• RQ1 How can pedestrian behavior in high-density regions be modeled to allow for both avoidance and navigation, rather than assuming universal avoidance?
• RQ2 What is the mathematical structure of the generalized Hughes model as a non-separable mean field game system, and how does it differ from classical formulations?
• RQ3 Under what conditions do weak solutions exist for the generalized Hughes model, and how do they behave under the vanishing viscosity limit?
• RQ4 How does the parameter β influence the emergent crowd dynamics, and what are the implications for evacuation and safety planning?
• RQ5 What numerical instabilities or concentration phenomena arise in the model, and under what parameter regimes do they occur?

Key findings

• Weak solutions exist for the generalized Hughes model under appropriate regularity and approximation assumptions, established via a viscosity-regularized approximation scheme.
• The vanishing viscosity limit of weak solutions is analyzed, showing convergence to a solution of the original non-regularized system, which supports the robustness of the model.
• Numerical simulations confirm that β = 2 leads to effective avoidance of high-density regions, resulting in dispersion and diffusion over time, consistent with intelligent crowd behavior.
• For β = 1, pedestrians are indifferent to density, leading to uniform flow toward exits, resembling classical Hughes dynamics with no avoidance behavior.
• When β = 0, pedestrians are attracted to high-density regions, causing concentration phenomena and numerical instability beyond T = 13.5, indicating potential model breakdown under strong attraction.
• The parameter β acts as a tunable safety control mechanism, enabling the model to simulate diverse evacuation scenarios and support real-time crowd management strategies.

Figure 2: Schematic illustration of density contour levels (a) and velocity field (b) at final time $\displaystyle T$ and $\displaystyle\beta=2$ .