Network formation as the fingering instability and its inverse problem

Many natural growth processes can be described as a Stefan problem, where the boundary between two phases can move with time. A specific example of such growth is Saffman-Taylor instability, and speaking more generally, Laplacian growth, in which the interface moves with the speed proportional to the gradient of a harmonic field. The growing interface is in general unstable to perturbations that evolve into fingers. The fingers frequently split as they grow, with the daughter branches competing with each other for the available flux. This results in a formation of a ramified, network-like pattern.

In our study, we take a limit of infinitely thin fingers, which grow only at their tips. Such description can be used to understand the formation of blood vessels, river networks, leaf venation or dielectric breakdown patterns, only to mention a few.

We then formulate the inverse problem – can the growth dynamics be inferred from the analysis of the final geometrical structure of the network? We approach this problem numerically, developing tools to simulate the growth and to extract growth laws of the network based on the final pattern. The reliability of the method is validated against synthetic data.