In generative models, Optimal Transport (OT) is used to improve Flow Matching (FM) by reducing noise-data coupling cost. However, different noise-to-output assignments can yield nearly equal costs, raising a key question. Is cost alone sufficient to guide coupling design? We address this question by separating transport cost from routing, i.e., the destination reached by each noise sample. We show numerically how FM and OT can differ in routing while remaining close in cost. We examine its consequences in learned neural FM. Using the exact FM routing as an oracle, we further construct a routing-aware training coupling and find that it yields a directionally consistent improvement in generation over a cost-matched, cost-only counterpart. Our findings highlight what cost minimization can overlook and motivate using both cost and routing to evaluate the design of OT-based FM couplings. Code will be released upon acceptance.
Figures & tables
Figure 1 : Similar transport cost but different routing between FM and OT. Noise samples in each region arrive at the same target atom in the data space.
Figure 2 : Different routing-boundary responses of FM and OT to the same atom rotation. Red: Exact FM, Blue: OT.
Quantity
Metric
Theory
Measured
Displacement
R2 , full grid
–
0.9986
R2 , ∣θ∣≤2∘
–
0.9991
Sign change at β∗
0.6455
0.6455
Routing mass
Log–log slope
1
0.9992
A vs. D/∣θ∣
0.038038
0.038040
Excess cost
Log–log slope
2
1.9976
Table 1 : Numerical validation of the boundary-displacement prediction and the resulting cost–routing separation in Eq. 18 .