Inverse Optimal Transport Modelling of Hidden Barriers to Secondary-School Access in Dispersed Rural Settlements

Distance is observable, but the effective cost of secondary-school access in dispersed rural settlements is shaped by barriers that are partly hidden in enrolment and attendance flows. This study develops an entropic inverse optimal transport framework that recovers latent generalized access costs from origin–school flow matrices while respecting learner totals and school capacities. The model combines network distance with river crossings, road condition, safety exposure, transport fares and household–school compatibility. Existence, uniqueness, Sinkhorn representation, invariance, parameter identifiability, stability and accessibility bounds are established. A reproducible synthetic experiment with 24 settlements, five schools and 1,230 capacity-weighted places demonstrates the method without representing simulated observations as field evidence. The full inverse model reduced Kullback–Leibler flow error from 0.38579 for a distance-only specification to 0.00431 and recovered standardized barrier coefficients with an Euclidean error of 0.3157. River crossings and transport fares produced the largest non-distance penalties. A soft-min accessibility index translated the recovered cost surface into settlement priorities. Counterfactual transport subsidies and a hostel reduced both average and upper-tail generalized costs, whereas a candidate school-placement rule exposed an equity efficiency trade-off. The framework therefore converts routinely available origin–destination counts into interpretable planning evidence, while explicitly separating structural inference from causal claims. Field validation, uncertainty quantification and participatory specification remain necessary before operational allocation.