  We consider the problem of understanding the coordinated movements of
biological or artificial swarms. In this regard, we propose a learning scheme
to estimate the coordination laws of the interacting agents from observations
of the swarm's density over time. We describe the dynamics of the swarm based
on pairwise interactions according to a Cucker-Smale flocking model, and
express the swarm's density evolution as the solution to a system of mean-field
hydrodynamic equations. We propose a new family of parametric functions to
model the pairwise interactions, which allows for the mean-field macroscopic
system of integro-differential equations to be efficiently solved as an
augmented system of PDEs. Finally, we incorporate the augmented system in an
iterative optimization scheme to learn the dynamics of the interacting agents
from observations of the swarm's density evolution over time. The results of
this work can offer an alternative approach to study how animal flocks
coordinate, create new control schemes for large networked systems, and serve
as a central part of defense mechanisms against adversarial drone attacks.
