Stochastic Gradient Descent for Traffic and Pedestrian Models via Neural Networks and Parameter Identification

Abstract

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.