p-Wasserstein distances on networks and 3D to 1D convergence

We study transport distances on metric graphs representing gas networks. Starting from the dynamic formulation of the Wasserstein distance, we review extensions to networks, with and without the possibility of storing mass on the vertices. Next, we examine the asymptotic behavior of the static Wasserstein distance on a three-dimensional network domain that converges to a metric graph. We show convergence of the distance with a proof that is based on the characterization of optimal transport plans as $c$-cyclically monotone sets. We conclude by illustrating our finding with several numerical examples.

💡 Research Summary

This paper develops a comprehensive framework for optimal transport on metric graphs, motivated by the modeling of gas pipeline networks. Starting from the classical Monge–Kantorovich formulation of the p‑Wasserstein distance, the authors replace the Euclidean ground cost with the shortest‑path distance on a metric graph, thereby defining a static p‑Wasserstein distance that respects the network topology. They then extend the dynamic Benamou–Brenier formulation to such graphs. The continuity equation is written on each edge, and two distinct vertex coupling conditions are considered. The first is the classical Kirchhoff condition, which enforces that the sum of incoming and outgoing fluxes at a node vanishes, implying no mass storage at vertices. The second introduces a vertex mass variable γᵥ(t) and a corresponding balance law ∂ₜγᵥ(t)=∑ₑ∈E(v) ĵₑᵥ(t), allowing mass to be stored at junctions. In both cases the flux is expressed as the momentum Jₜ=ρₜvₜ, leading to a linear continuity equation and a convex dynamic optimal transport problem.

A major contribution is the rigorous analysis of the limit where a three‑dimensional pipe network Ω_ε, with pipe radius ε, collapses onto a one‑dimensional metric graph G as ε→0. The authors prove that the static p‑Wasserstein distance on Ω_ε converges to the graph‑based distance on G. The proof hinges on the c‑cyclical monotonicity of optimal transport plans: optimal couplings π_ε for the 3D problem satisfy a monotonicity condition with respect to the Euclidean cost |x−y|ᵖ, which, after scaling, converges to the graph cost d_G(x,y)ᵖ. Using Γ‑convergence of the associated transport functionals and tightness arguments for the measures, they establish the convergence Wₚ(μ_ε,ν_ε)→Wₚ(μ,ν). This result provides a solid mathematical justification for replacing detailed 3D pipe models by 1D graph models in applications.

The theoretical findings are illustrated through several numerical experiments. The authors compute optimal transport plans on (i) a graph with Kirchhoff coupling, (ii) a graph with vertex storage, and (iii) the full 3D pipe geometry discretized by finite elements. The static distances are evaluated using the Sinkhorn algorithm, while the dynamic flows are obtained via a finite‑element discretization of the continuity equation. The simulations confirm that, for sufficiently small ε, the 3D and 1D models yield virtually identical distances and flow patterns, and they highlight the flexibility of the storage model compared to the pure Kirchhoff case.

Overall, the paper makes three key advances: (1) it formalizes dynamic p‑Wasserstein transport on metric graphs with both Kirchhoff and storage vertex conditions; (2) it provides a rigorous 3D‑to‑1D convergence proof based on c‑cyclical monotonicity and Γ‑convergence; and (3) it validates the theory with concrete numerical examples. These contributions lay a solid foundation for the optimal design, control, and analysis of complex networked systems such as gas, water, or electricity distribution networks.