Branching structures emerging from a continuous optimal transport model

Highlights: • Dynamic formulation of ramified transport problems is solved via standard FEM and time stepping methods. • Singular structures (complex networks) emerge in a continuous setting. • Singular structures correspond to minima of a Lyapunov function closely related to more standard branched transport models. • Numerical solution of divergence constraint minimization problems in multi-dimensions. Recently a Dynamic-Monge-Kantorovich formulation of the PDE-based -optimal transport problem was presented. The model considers a diffusion equation enforcing the balance of the transported masses with a time-varying conductivity that evolves proportionally to the transported flux. In this paper we present an extension of this model that considers a time derivative of the conductivity that grows as a power-law of the transport flux with exponent $\beta >0$. A sub-linear growth ($0<\beta <1$) penalizes the flux intensity and promotes distributed transport, with equilibrium solutions that are reminiscent of Congested Transport Problems. On the contrary, a super-linear growth ($\beta >1$) favors flux intensity and promotes concentrated transport, leading to the emergence of steady-state "singular" and "fractal-like" configurations that resemble those of Branched Transport Problems. We derive a numerical discretization of the proposed model that is accurate, efficient, and robust for a wide range of scenarios. For $\beta >1$ the numerical model is able to reproduce highly irregular and fractal-like formations without any a-priory structural assumption.