The inverse problem in fingered growth – inferring growth rules from the final pattern.

Many physical processes involving moving boundaries produce fingering patterns. Well-known examples include viscous fingering, electrochemical deposition or crystallization in supercooled liquids. They all involve a field (pressure, concentration or temperature depending on a problem) which is driving the growth, with the growth velocity dependent on the field gradient.  The growing interface is often unstable to perturbations that evolve into fingers. The fingers 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 this study we consider the inverse problem for the fingered growth - can the growth dynamics be inferred from the analysis of the final geometrical structure of the network? We focus on the case where the fingers are thin, growing only at the tips, which considerably simplifies the mathematical description. We show that by a combination of backward and forward evolution of a network one not only can reconstruct the growth rules but also get an insight into the conditions under which the finger splitting occurs. Finally, we apply the model to analyze the growth of the river network system