Technical Summary: Learning Urban Access Costs from Origin–Destination Flows via Inverse Optimal Transport

Problem Statement

Urban planners frequently manage mixed public-private service networks (e.g., education, healthcare, transit) where households make choices based on a latent cost function balancing distance, price, and institutional quality. While planners can observe the resulting origin-destination (OD) flows, they lack direct visibility into the behavioral cost surface driving these decisions.

This paper addresses this gap using the context of school choice in the Philippines. Specifically, it examines the country's largest national education subsidy, which funds enrollment in private junior high schools to alleviate congestion in public schools. The core planning challenge is determining the "spatial reach" of such a subsidy: How many kilometers of perceived travel cost does a subsidy offset, and does this value vary across different spatial regimes? Traditional forward models (e.g., gravity or discrete-choice models) are effective for prediction but fail to directly recover the underlying cost function necessary for answering these planning questions.

Methodology

The authors frame the problem as Inverse Optimal Transport (OT). Given an observed flow matrix $T_{obs}$ (normalized as a transport plan with marginals $p$ and $q$), the goal is to recover the cost matrix $C$ that makes the observed flows most plausible under an entropic OT model.

The Forward Model

The forward process models the flow $\hat{T}$ as the solution to an entropic optimal transport problem:
$$\hat{T} = \text{Sinkhorn}(p, q, C, \varepsilon)$$
where $\varepsilon = 0.1$ is the regularization parameter.

The Inverse Problem

The objective is to estimate the cost function parameters that minimize the discrepancy between observed and predicted flows:
$$\Phi = \frac{\|T_{obs} - \hat{T}\|_F^2}{2\sigma^2}$$

To solve this, the paper proposes two complementary cost models:

1. Interpretable Distance-Banded Model:
   This parametric model defines the cost $c_{ij}$ between origin $i$ and destination $j$ as:
   $$c_{ij} = \theta(k(d_{ij})) \cdot d_{ij} + \theta_e e_j$$

   • $d_{ij}$: Road-network distance.
   • $e_j$: Destination subsidy amount (in thousands of pesos).
   • $k(d_{ij})$: An index for distance bands (0–5 km, 5–15 km, 15–50 km).
   • Key Metric: The model derives the subsidy-equivalent distance $\lambda(k) = -\frac{\theta_e}{\theta_d(k)}$, representing the kilometers of perceived travel cost offset by a 1,000-peso subsidy increase in a specific band.

2. Neural Cost Model:
   A non-parametric approach where the cost is a neural network function $c_{ij} = f_w(d_{ij}, e_j)$. This model is trained by backpropagating through a differentiable Sinkhorn operator, allowing the network to learn complex, non-linear interactions between distance and subsidies without pre-defined bands.

Data

The study utilizes administrative data from the Philippine Department of Education for the 2022–2024 school years in the most populous region.

• Scale: 4,200 origin schools, 1,861 destination schools.
• Flows: 23,820 observed positive OD pairs representing 283,016 learner trips.
• Constraints: Only feasible pairs within 50 km (1,852,177 pairs) were considered. All data is aggregated; no individual learner is identifiable.

Key Results

1. Performance Comparison

• Piecewise Model: Using GPU-accelerated Maximum A Posteriori (MAP) optimization, the piecewise model reduced the loss $\Phi$ from an initial 5.00 to 3.29.
• Neural Model: The neural inverse OT model achieved a lower loss of 2.84, representing a 13.7% improvement in fit over the piecewise model.

2. Subsidy-Equivalent Distance ($\lambda$)

The piecewise model yielded the following estimates for the kilometers of perceived cost offset per 1,000 pesos:

• 0–5 km band: $\lambda(1) = 5.35$ km
• 5–15 km band: $\lambda(2) = 6.07$ km
• 15–50 km band: $\lambda(3) = 1.00$ km

Interpretation: In the 5–15 km band, a 1,000-peso subsidy increase makes a school appear roughly 6.07 km closer in terms of perceived travel cost. This is a change in the choice-cost function, not a reduction in physical travel distance.

3. Distance-Subsidy Interactions

Analysis of the neural model's distance derivative ($dc/dd$) revealed nuances missed by the piecewise approach:

• At zero subsidy, sensitivity is smooth and low (0.083–0.196).
• At median (9,000 pesos) and maximum (13,000 pesos) subsidies, sensitivity increases significantly at short distances but flattens over medium and long ranges.
• The neural model suggests that the sharp increase in friction implied by the 15–50 km band in the piecewise model may be an artifact of band aggregation rather than a true behavioral threshold.

Significance and Claims

The paper claims three primary contributions to urban planning and computational social science:

1. Methodological Innovation: It introduces a dual-model framework for inverse OT, combining an interpretable piecewise estimator with a flexible neural estimator trained via differentiable Sinkhorn.
2. Policy-Relevant Metrics: It derives the subsidy-equivalent distance $\lambda(k)$, a metric that translates abstract subsidy amounts into tangible spatial accessibility gains. This allows planners to calibrate subsidies and site facilities based on how much "perceived distance" a subsidy effectively buys.
3. Generalizability: While demonstrated on Philippine school choice, the authors assert the framework applies to any urban system involving mixed networks and OD flows, including health facility choice, transit ridership, subsidized housing, and evacuation behavior.

The central planning implication is that subsidies possess a spatial footprint. Their effectiveness is not uniform; it depends on the alignment between subsidized facility locations and demand, as well as the distance sensitivity of households in specific network regions. The framework provides a tool to transform administrative flow data into actionable metrics for accessibility-aware service allocation.