  The aim of this paper is to solve an inverse problem which regards a mass
moving in a bounded domain. We assume that the mass moves following an unknown
velocity field and that the evolution of the mass density can be described by
partial differential equations (PDEs), which is also unknown. The input data of
the problems are given by some snapshots of the mass distribution at certain
times, while the sought output is the velocity field that drives the mass along
its displacement. To this aim, we put in place an algorithm based on the
combination of two methods: first, we use the Dynamic Mode Decomposition to
create a mathematical model describing the mass transfer; second, we use the
notion of Wasserstein distance (also known as earth mover's distance) to
reconstruct the underlying velocity field that is responsible for the
displacement. Finally, we consider a real-life application: the algorithm is
employed to study the travel flows of people in large populated areas using, as
input data, density profiles (i.e. the spatial distribution) of people in given
areas at different time instances. This kind of data are provided by the
Italian telecommunication company TIM and are derived by mobile phone usage.
