Online coordinated estimation of origin–destination, path, and link flows in urban rail transit networks via a physically constrained computational graph

Urban rail transit (URT) systems require timely estimates of origin–destination (OD) demand and network-wide link flows to support crowding management and disruption response under rapidly changing conditions. Despite the growing availability of multisource data, online estimation remains a challenging problem because OD demand is latent and high-dimensional, passenger propagation is time-dependent, observations are often sparse or incomplete, and the estimated states must remain physically consistent across OD, path, and link levels. Existing studies often focus on OD inference or link-flow estimation separately, or use static path-to-link assignment that cannot adapt to time-varying service and congestion conditions. This paper develops an online coordinated estimation framework for URT networks based on a physically constrained computational graph. The framework jointly estimates OD, path, and link flows within a unified time-expanded structure, rather than treating them as separate outputs. A time-dependent path-to-link assignment matrix is introduced to represent dynamic passenger propagation under changing congestion conditions. Physical and operational constraints are embedded directly in the graph to preserve consistency and interpretability. A rolling-horizon recalibration mechanism further allows later observations to revise earlier estimates across intervals. By integrating model-based priors with real-time observations, the framework combines structural interpretability with data-driven adaptability under sparse and incomplete measurements. Experiments on the Sioux Falls network and the real-world Beijing metro network demonstrate that the proposed framework provides accurate and stable estimates of OD demand, path flows, and link flows at the network scale. Comparisons with different baselines show that the framework is particularly advantageous under sparse or incomplete observations, especially when the task extends from OD inference alone to native, physically consistent network-wide link-flow estimation.