Geometric flows of branched transportation networks
Abstract
The branched transport problem is a nonconvex and nonsmooth variational optimization problem on normal $1$-currents in ℝn with prescribed boundary. The optimality is with respect to some non-decreasing, lower semicontinuous, and subadditive function τ:ℝ+→ℝ+ with τ(0)=0 describing the cost τ(m) to move an amount of mass m per unit distance. The subadditivity leads to complicated, hierarchically ramified patterns in the support of (suboptimal) solutions. These network-like sets appear to have regularity properties similar to those of the singular surfaces arising in the so-called Brakke flow, a weak generalization of the mean curvature flow. We construct a geometric flow of transportation networks, which correspond to normal real $1$-rectifiable currents, by modifying Brakke's variational approximation scheme. We prove the existence of a limit that is Hölder continuous with respect to the flat norm and whose branched transport cost decreases along the geometric evolution. We further analyze a closely related geometric flow approximated by $1$-varifolds, whose weight measures model the branched transport cost, and establish its $1$-rectifiability together with a motion law analogous to Brakke's inequality.
Related
- SIGMA: Symmetry-aware, Intelligent, Geometric, Multi-faced Adaptive Traffic Control for Reliable Intelligent Transportation Systems
- On the Observability and Redundancy of Intelligent Transportation Networks
- Route optimization for urban hazardous materials transportation under time-varying road networks
- Keyword-Coverage Optimal Route Search on Temporal Transportation Networks
- Spontaneous Geometric Structures in 2D MHD Flows
BibTeX
@misc{lohmann2026geometric,
title = {{Geometric flows of branched transportation networks}},
author = {Julius Lohmann and Yoshihiro Tonegawa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24451},
eprint = {2609.24451},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24451},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometric flows of branched transportation networks
AU - Julius Lohmann
AU - Yoshihiro Tonegawa
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24451
UR - https://arxiv.org/abs/2609.24451
AB - The branched transport problem is a nonconvex and nonsmooth variational optimization problem on normal $1$-currents in ℝ^n with prescribed boundary. The optimality is with respect to some non-decreasing, lower semicontinuous, and subadditive function τ:ℝ_+→ℝ_+ with τ(0)=0 describing the cost τ(m) to move an amount of mass m per unit distance. The subadditivity leads to complicated, hierarchically ramified patterns in the support of (suboptimal) solutions. These network-like sets appear to have regularity properties similar to those of the singular surfaces arising in the so-called Brakke flow, a weak generalization of the mean curvature flow. We construct a geometric flow of transportation networks, which correspond to normal real $1$-rectifiable currents, by modifying Brakke's variational approximation scheme. We prove the existence of a limit that is Hölder continuous with respect to the flat norm and whose branched transport cost decreases along the geometric evolution. We further analyze a closely related geometric flow approximated by $1$-varifolds, whose weight measures model the branched transport cost, and establish its $1$-rectifiability together with a motion law analogous to Brakke's inequality.
ER -