Understanding how individual training observations influence model predictions is important for developing reliable and interpretable data-driven transportation systems.
Existing influence-function methods, however, are primarily formulated for unconstrained learning and do not explicitly account for physical constraints that couple different components of a transportation network.
Under such constraints, perturbing a training observation can affect model parameters and predictions only through physically feasible directions, potentially changing how its influence propagates across the network.
This paper develops a hard physics-informed influence-function framework for constrained learning, in which training-data influence is characterized within the physically feasible space defined by network constraints.
The analytical results reveal how physical constraints alter the magnitude, direction, and spatial distribution of training-sample influence relative to conventional influence measures.
Using traffic prediction under flow-conservation constraints as a case study, we show that influential observations identified by the proposed framework differ systematically from those identified by classical influence functions.
Through orthogonal decomposition, we quantitatively characterize how much of a perturbation in the training data is propagated through the network via the flow-conservation constraints and how much is removed by the constraint projection.
The proposed influence framework provides a physically consistent approach for evaluating training-data influence in constrained transportation learning problems, with potential applications to data diagnostics, sensor evaluation, and targeted data collection.