Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Martin Burger, Ariane Fazeny, Gilles Mordant, Jan-Frederik Pietschmann

TL;DR

The paper develops a rigorous framework for p-Wasserstein distances on networks by formulating dynamic transport on metric graphs with two node-coupling schemes (mass storage and Kirchhoff) and by connecting these to gradient-flow structures. It then analyzes a zero-width limit, proving convergence of 3D network OT costs to the 1D graph limit under atomless limits, and establishes conditions for the convergence of optimal transport maps along with space-time interpretations and topology-stability results. A key theoretical contribution is the convergence of OT as the network widens from 3D tubes to a 1D graph, supported by Kantorovich duality and cyclical monotonicity arguments, plus numerical demonstrations illustrating geodesic behavior on networks. The work provides a principled bridge between 3D gas-network models and efficient graph-based transport computations, with implications for network design, stability, and control under topological changes.

Abstract

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.

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

TL;DR

The paper develops a rigorous framework for p-Wasserstein distances on networks by formulating dynamic transport on metric graphs with two node-coupling schemes (mass storage and Kirchhoff) and by connecting these to gradient-flow structures. It then analyzes a zero-width limit, proving convergence of 3D network OT costs to the 1D graph limit under atomless limits, and establishes conditions for the convergence of optimal transport maps along with space-time interpretations and topology-stability results. A key theoretical contribution is the convergence of OT as the network widens from 3D tubes to a 1D graph, supported by Kantorovich duality and cyclical monotonicity arguments, plus numerical demonstrations illustrating geodesic behavior on networks. The work provides a principled bridge between 3D gas-network models and efficient graph-based transport computations, with implications for network design, stability, and control under topological changes.

Abstract

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 -cyclically monotone sets. We conclude by illustrating our finding with several numerical examples.
Paper Structure (21 sections, 7 theorems, 87 equations, 1 figure)

This paper contains 21 sections, 7 theorems, 87 equations, 1 figure.

Table of Contents

  1. Dynamic transport distances on metric graphs
  2. Metric graphs
  3. Continuity equations
  4. Mass storage on nodes
  5. Kirchhoff case
  6. Action functionals and dynamic distances
  7. Mass storage on nodes
  8. Kirchhoff case
  9. Gradient flows on metric graphs
  10. Mass storage on nodes
  11. Kirchhoff case
  12. Zero-width limit of gas networks
  13. 3D modeling of gas network
  14. Main results
  15. Convergence on the entire network.
  16. ...and 6 more sections

Key Result

lemma thmcounterlemma

Let $(\mu,{\mathrm j})\in{\mathsf{CE}}$ with then there exists ${\left({\tilde{\mu},\tilde{\mathbin{\clipbox{-.5 -.5 0 {.35}}{$ j$}\mspace{1mu}}}}\right)}\in{\mathsf{CE}}$ such that ${\left({\tilde{\mu}_t}\right)}_{t \in {\left[{0, 1}\right]}} \subset {\mathcal{P}\!\left({{\mathsf M}}\right)}$ is an absolutely continuous curve equipped with th is absolutely continuous.

Figures (1)

  • Figure 1.1: Example of estimated trajectories based on a space-discretized version of the problem. The source measure is the set of points in the lower left corner, while the target measure are the two accumulations of points at the right hand side.

Theorems & Definitions (25)

  • definition thmcounterdefinition: 1D distance on metric graph
  • definition thmcounterdefinition: Continuity equation, Heinze2024
  • lemma thmcounterlemma: Well-posedness of ${\mathsf{CE}}$, Heinze2024
  • proof
  • definition thmcounterdefinition: Continuity equation (Kirchhoff) erbar2021gradient
  • remark thmcounterremark
  • definition thmcounterdefinition: Absolutely continuous curve ambrosio2005gradient
  • definition thmcounterdefinition: Metric derivative ambrosio2005gradient
  • theorem 1: Absolutely continuous curves
  • remark thmcounterremark
  • ...and 15 more