  We show how a high-performing, fully distributed and symmetric neural
V-formation controller can be synthesized from a Centralized MPC (Model
Predictive Control) controller using Deep Learning. This result is significant
as we also establish that under very reasonable conditions, it is impossible to
achieve V-formation using a deterministic, distributed, and symmetric
controller. The learning process we use for the neural V-formation controller
is significantly enhanced by CEGkR, a Counterexample-Guided k-fold Retraining
technique we introduce, which extends prior work in this direction in important
ways. Our experimental results show that our neural V-formation controller
generalizes to a significantly larger number of agents than for which it was
trained (from 7 to 15), and exhibits substantial speedup over the MPC-based
controller. We use a form of statistical model checking to compute confidence
intervals for our neural V-formation controller's convergence rate and time to
convergence.
