Structure of Optimal Transport Networks Subject to a Global Constraint

In this work, we present a novel tool for reconstructing networks from corrupted images. The reconstructed network is the result of a minimization problem that has a misfit term with respect to the observed data, and a physics-based regularizing term coming from the theory of optimal transport. Through a range of numerical tests, we demonstrate that our suggested approach can effectively rebuild the primary features of damaged networks, even when artifacts are present.

While many large infrastructure networks, such as power, water, and natural gas systems, have similar physical properties governing flows, these systems tend to have distinctly different sizes and topological structures. This paper seeks to understand how these different size-scales and topological features can emerge from relatively simple design principles. Specifically, we seek to describe the conditions under which it is optimal to build decentralized network infrastructu...

For the problem of efficiently supplying material to a spatial region from a single source, we present a simple scaling argument based on branching network volume minimization that identifies limits to the scaling of sink density. We discuss implications for two fundamental and unresolved problems in organismal biology and geomorphology: how basal metabolism scales with body size for homeotherms and the scaling of drainage basin shape on eroding landscapes.

Highly-optimized complex transport networks serve crucial functions in many man-made and natural systems such as power grids and plant or animal vasculature. Often, the relevant optimization functional is non-convex and characterized by many local extrema. In general, finding the global, or nearly global optimum is difficult. In biological systems, it is believed that natural selection slowly guides the network towards an optimized state. However, general coarse grained model...

Optimal transport aims to learn a mapping of sources to targets by minimizing the cost, which is typically defined as a function of distance. The solution to this problem consists of straight line segments optimally connecting sources to targets, and it does not exhibit branching. These optimal solutions are in stark contrast with both natural, and man-made transportation networks, where branching structures are prevalent. Here we discuss a fast heuristic branching method for...

Inspired by the matching of supply to demand in logistical problems, the optimal transport (or Monge--Kantorovich) problem involves the matching of probability distributions defined over a geometric domain such as a surface or manifold. In its most obvious discretization, optimal transport becomes a large-scale linear program, which typically is infeasible to solve efficiently on triangle meshes, graphs, point clouds, and other domains encountered in graphics and machine lear...

Flow networks efficiently transport nutrients and other solutes in many physical systems, such as plant and animal vasculature. In the case of the animal circulatory system, an adequate oxygen and nutrient supply is not guaranteed everywhere: as nutrients travel through the microcirculation and get absorbed, they become less available at the venous side of the vascular network. Ensuring that the nutrient distribution is homogeneous provides a fitness advantage, as all tissue ...

This paper presents a Wasserstein attraction approach for solving dynamic mass transport problems over networks. In the transport problem over networks, we start with a distribution over the set of nodes that needs to be "transported" to a target distribution accounting for the network topology. We exploit the specific structure of the problem, characterized by the computation of implicit gradient steps, and formulate an approach based on discretized flows. As a result, our p...

In this article we survey recent progress on mathematical results on gas flow in pipe networks with a special focus on questions of control and stabilization. We briefly present the modeling of gas flow and coupling conditions for flow through vertices of a network. Our main focus is on gas models for spatially one-dimensional flow governed by hyperbolic balance laws. We survey results on classical solutions as well as weak solutions. We present results on well--posedness...

This article deals with the problem of finding the best topology, pipe diameter choices, and operation parameters for realistic district heating networks. Present design tools that employ non-linear flow and heat transport models for topological design are limited to small heating networks with up to 20 potential consumers. We introduce an alternative adjoint-based numerical optimization strategy to enable large-scale nonlinear thermal network optimization. In order to avoid ...