Biological transportation networks are fundamental building blocks of nature. Typical examples are leaf venation, angiogenesis and neural networks. Mathematical modeling efforts so far have been mostly based on discrete graph-based methods (deterministic or random) and optimal mass transportation techniques relying on the Monge-Kantorovich approach. A new bottom-up modeling approach has been designed recently by D. Cai et al, based on first physical principles, and analysed and implemented by P. Markowich, B. Perthame et al.
This lead to a very complex nonlinear system of partial differential equations with highly singular scaling limits and many still open mathematical questions (strong solutions, uniqueness, regularity, measure valued limits concentrating on sets of Hausdorff dimension one, just to name a few). The curse of ‘large data’ has to be overcome when efficient numerical algorithms are sought, particularly considering the necessity to resolve network fine-structures in the context of stable and accurate numerical methods as well as when associated inverse problems are considered. In particular, stability and efficiency of networks cannot be measured anymore in terms of classical PDE concepts but statistical methods have to be used instead.
We believe that very similar modeling, analysis and numerical techniques can be applied to networks in other applied sciences, like information transport networks in social sciences.