发表机构
San Diego State University（圣地亚哥州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对交通与行人动力学的ODE模型，结合最优控制与随机梯度下降，提出两种二阶模型，经数值实验验证神经网络行人模型成本最低，二阶模型可实现个性化可解释动力学。
AI 中文摘要
本研究通过最优控制与随机梯度下降法，对交通与行人动力学中由相互作用驱动的常微分方程（ODE）模型进行校准。相互作用力要么由经典的受物理启发的定律规定，其参数从轨迹数据中辨识得到；要么由前馈神经网络表示，其权重作为控制变量。对于神经网络驱动的粒子系统，我们证明了状态方程的适定性、轨迹对初始数据及网络参数的Lipschitz连续依赖性、最优参数集的存在性，以及带有显式伴随方程的一阶最优性系统。所得的梯度伴随表示与投影小批量ADADELTA方案相结合，使得每次梯度评估仅需一次正向和一次反向ODE求解，与网络参数数量无关。基于该框架，我们提出了两种新的二阶模型，分别用于车辆交通和人群，它们结合了目标追寻、学习到的相互作用、排斥-对齐或耗散以及外部控制项。数值实验从成本降低和轨迹均方根偏差两个方面比较了六种模型：Lighthill-Whitham-Richards型的跟驰模型、神经网络交通模型、社会力模型、神经网络行人模型，以及两种新模型。神经网络行人模型实现了最低的最终成本，而更复杂的二阶模型则以更高的校准成本换取个性化、行为可解释的动力学。最后，我们讨论了研究的局限性以及纳入更丰富的行为和环境效应的方向。
英文摘要
We study the calibration of interaction-driven ordinary differential equation (ODE) models for traffic and pedestrian dynamics by means of optimal control and stochastic gradient descent. The interaction forces are either prescribed by classical physics-inspired laws, whose parameters are identified from trajectory data, or represented by feed-forward neural networks whose weights act as control variables. For the neural-network-driven particle system we prove well-posedness of the state equation, Lipschitz-continuous dependence of the trajectories on the initial data and on the network parameters, existence of an optimal parameter set, and a first-order optimality system with an explicit backward adjoint equation. The resulting adjoint representation of the gradient is combined with a projected mini-batch ADADELTA scheme, so that each gradient evaluation requires one forward and one backward ODE solve, independently of the number of network parameters. Building on this framework we propose two new second-order models, one for vehicular traffic and one for crowds, that combine goal-seeking, learned interaction, repulsion-alignment or dissipation, and external control terms. Numerical experiments compare six models (a follow-the-leader model of Lighthill-Whitham-Richards type, a neural-network traffic model, the social force model, a neural-network pedestrian model, and the two new models) in terms of cost reduction and root-mean-square trajectory deviation. The neural-network pedestrian model achieves the lowest final cost, while the richer second-order models trade a higher calibration cost for individualised, behaviourally interpretable dynamics. We conclude by discussing limitations and directions for incorporating richer behavioural and environmental effects.
Comments31 pages, 8 figures
