  In this work, we study the inverse problem of identifying complex flocking
dynamics in a domain cluttered with obstacles. We get inspiration from animal
flocks moving in complex ways with capabilities far beyond what current robots
can do. Owing to the difficulty of observing and recovering the trajectories of
the agents, we focus on the dynamics of their probability densities, which are
governed by partial differential equations (PDEs), namely compressible Euler
equations subject to non-local forces. We formulate the inverse problem of
learning interactions as a PDE-constrained optimization problem of minimizing
the squared Hellinger distance between the histogram of the flock and the
distribution associated to our PDEs. The numerical methods used to efficiently
solve the PDE-constrained optimization problem are described. Realistic
flocking data are simulated using the Boids model of flocking agents, which
differs in nature from the reconstruction models used in our PDEs. Our analysis
and simulated experiments show that the behavior of cohesive flocks can be
recovered accurately with approximate PDE solutions.
