A paradigm put forth by E. Schr\"odinger in 1931/32, known as Schr\"odinger bridges, represents a formalism to pose and solve control and estimation problems seeking a perturbation from an initial control schedule (in the case of control), or from a prior probability law (in the case of estimation), sufficient to reconcile data in the form of marginal distributions and minimal in the sense of relative entropy to the prior. In the same spirit, we consider traffic-flow and apply a Schr\"odinger-type dictum, to perturb minimally with respect to a suitable relative entropy functional a prior schedule/law so as to reconcile the traffic flow with scarce aggregate distributions on families of indistinguishable individuals. Specifically, we consider the problem to regulate/estimate multi-commodity network flow rates based only on empirical distributions of commodities being transported (e.g., types of vehicles through a network, in motion) at two given times. Thus, building on Schr\"odinger's large deviation rationale, we develop a method to identify {\em the most likely flow rates (traffic flow)}, given prior information and aggregate observations. Our method further extends the Schr\"odinger bridge formalism to the multi-commodity setting, allowing commodities to exit or enter the flow field as well (e.g., vehicles to enter and stop and park) at any time. The behavior of entering or exiting the flow field, by commodities or vehicles, is modeled by a Markov chains with killing and creation states. Our method is illustrated with a numerical experiment.

翻译：E. Schrödinger于1931/32年提出的范式——即薛定谔桥——提供了一种形式体系，用于提出并解决控制与估计问题：在控制问题中寻求对初始控制方案的扰动，在估计问题中寻求对先验概率律的扰动，这种扰动足以协调以边际分布形式给出的数据，且在相对熵意义上相对于先验是最小的。秉承相同的精神，我们考虑交通流问题，并应用薛定谔类型的准则，在合适的相对熵泛函意义下对先验调度/律进行最小扰动，以协调交通流与不可区分个体族群的稀缺聚合分布。具体而言，我们考虑仅基于两种给定时间点被运输商品（例如，网络中运动中的车辆类型）的经验分布，来调节/估计多商品网络流率的问题。因此，基于薛定谔的大偏差原理，我们发展了一种方法，在给定先验信息和聚合观测的条件下，识别最可能的流率（交通流）。我们的方法进一步将薛定谔桥形式体系扩展到多商品场景，允许商品在任何时刻退出或进入流场（例如，车辆进入、停止和停放）。商品或车辆进入或退出流场的行为，通过具有湮灭和生成状态的马尔可夫链建模。最后，我们通过数值实验对所提方法进行说明。