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This work studies a macroscopic traffic flow model driven by a system of nonlinear hyperbolic partial differential equations. Using Lie symmetry analysis, we determine the infinitesimal generators and construct an optimal system of…

Analysis of PDEs · Mathematics 2025-08-26 Urvashi Joshi , Aniruddha Kumar Sharma , Rajan Arora

We study statistical properties of a family of maps acting in the space of integer valued sequences, which model dynamics of simple deterministic traffic flows. We obtain asymptotic (as time goes to infinity) properties of trajectories of…

Dynamical Systems · Mathematics 2007-05-23 Michael Blank

We extend the Aw-Rascle macroscopic model of car traffic into a two-way multi-lane model of pedestrian traffic. Within this model, we propose a technique for the handling of the congestion constraint, i.e. the fact that the pedestrian…

Mathematical Physics · Physics 2014-04-08 Cécile Appert-Rolland , Pierre Degond , Sébastien Motsch

This paper firstly show that 2 Dimensional Intelligent Driver Model (Jiang et al., PloS one, 9(4), e94351, 2014) is not able to replicate the synchronized traffic flow. Then we propose an improved model by considering the difference between…

Pattern Formation and Solitons · Physics 2016-08-11 Junfang Tian , Rui Jiang , Geng Li , Martin Treiber , Chenqiang Zhu , Bin Jia

To perform autonomous driving maneuvers, such as parallel or perpendicular parking, a vehicle requires continual speed and steering adjustments to follow a generated path. In consequence, the path's quality is a limiting factor of the…

Systems and Control · Electrical Eng. & Systems 2025-05-14 Jason Zalev

Traditional methods for autonomous driving are implemented with many building blocks from perception, planning and control, making them difficult to generalize to varied scenarios due to complex assumptions and interdependencies. Recently,…

Computer Vision and Pattern Recognition · Computer Science 2020-11-03 Peide Cai , Yuxiang Sun , Hengli Wang , Ming Liu

The Gipps car-following model is a widely used tool for studying and simulation traffic dynamics. Despite its popularity an often disregarded property is that under heterogeneous parametrization on the individual vehicles in the traffic…

Adaptation and Self-Organizing Systems · Physics 2019-02-14 Leonhard Lücken

We introduce a general formulation for an implicit equation-free method in the setting of slow-fast systems. First, we give a rigorous convergence result for equation-free analysis showing that the implicitly defined coarse-level time…

Dynamical Systems · Mathematics 2015-08-03 Christian Marschler , Jan Sieber , Rainer Berkemer , Atsushi Kawamoto , Jens Starke

Flow-based methods for sampling and generative modeling use continuous-time dynamical systems to represent a {transport map} that pushes forward a source measure to a target measure. The introduction of a time axis provides considerable…

Machine Learning · Statistics 2025-06-19 Panos Tsimpos , Zhi Ren , Jakob Zech , Youssef Marzouk

We critically discuss the concept of ``synchronized flow'' from a historical, empirical, and theoretical perspective. Problems related to the measurement of vehicle data are highlighted, and questionable interpretations are identified.…

Statistical Mechanics · Physics 2007-05-23 D. Helbing , I. Farkas , D. Fasold , M. Treiber , T. Vicsek

The paper provides results for a non-standard, hyperbolic, 1-D, nonlinear traffic flow model on a bounded domain. The model consists of two first-order PDEs with a dynamic boundary condition that involves the time derivative of the…

Optimization and Control · Mathematics 2017-07-10 Iasson Karafyllis , Nikolaos Bekiaris-Liberis , Markos Papageorgiou

We discuss numerical strategies to deal with PDE systems describing traffic flows, taking into account a density threshold, which restricts the vehicles density in the situation of congestion. These models are obtained through asymptotic…

Numerical Analysis · Mathematics 2016-12-13 Florent Berthelin , Thierry Goudon , Bastien Polizzi , Magali Ribot

This paper proposes a novel class of data-driven acceleration methods for steady-state flow field solvers. The core innovation lies in predicting and assigning the asymptotic limit value for each parameter during iterations based on its own…

Fluid Dynamics · Physics 2025-07-08 Zikun Liu , Xukun Wang , Yilang Liu , Weiwei Zhang

The congestion formation on a urban road network is one of the key issue for the development of a sustainable mobility in the future smart cities. In this work we propose a reductionist approach studying the stationary states of a simple…

Physics and Society · Physics 2024-05-28 Lorenzo Di Meco , Mirko Degli Esposti , Federico Bellisardi , Armando Bazzani

Fast flow models accelerate the iterative sampling process by learning to directly predict ODE path integrals, enabling one-step or few-step generation. However, we argue that current fast-flow training paradigms suffer from two fundamental…

Computer Vision and Pattern Recognition · Computer Science 2026-02-09 Tianyi Zhang , Chengcheng Liu , Jinwei Chen , Chun-Le Guo , Chongyi Li , Ming-Ming Cheng , Bo Li , Peng-Tao Jiang

Optimal transport (OT) theory provides a principled framework for modeling mass movement in applications such as mobility, logistics, and economics. Classical formulations, however, generally ignore capacity limits that are intrinsic in…

Optimization and Control · Mathematics 2025-11-04 Anqi Dong , Karl Henrik Johansson , Johan Karlsson

In this paper a spatial homogeneous vehicular traffic flow model based on a stochastic master equation of Boltzmann type in the acceleration variable is solved numerically for a special driver interaction model. The solution is done by a…

Other Condensed Matter · Physics 2009-11-10 K. Thomas Waldeer

In this paper, we propose the ultra-discrete optimal velocity model, a cellular-automaton model for traffic flow, by applying the ultra-discrete method for the optimal velocity model. The optimal velocity model, defined by a differential…

Cellular Automata and Lattice Gases · Physics 2009-05-23 Masahiro Kanai , Shin Isojima , Katsuhiro Nishinari , Tetsuji Tokihiro

Data-driven modelling and synthesis of motion is an active research area with applications that include animation, games, and social robotics. This paper introduces a new class of probabilistic, generative, and controllable motion-data…

Machine Learning · Computer Science 2020-12-08 Gustav Eje Henter , Simon Alexanderson , Jonas Beskow

In this article, we investigate the long-term behavior of the ``Bando--follow-the-leader'' car-following model, whose well-posedness and stability with respect to delay were analyzed in a recent work \cite{gong2023well}. We first establish…

Classical Analysis and ODEs · Mathematics 2026-03-03 Fei Cao , Xiaoqian Gong , Alexander Keimer