  Modeling and control of agent-based models is twice cursed by the
dimensionality of the problem, as both the number of agents and their state
space dimension can be large. Even though the computational barrier posed by a
large ensemble of agents can be overcome through a mean field formulation of
the control problem, the feasibility of its solution is generally guaranteed
only for agents operating in low-dimensional spaces. To circumvent the
difficulty posed by the high dimensionality of the state space a kinetic model
is proposed, requiring the sampling of high-dimensional, two-agent
sub-problems, to evolve the agents' density using a Boltzmann type equation.
Such density evolution requires a high-frequency sampling of two-agent optimal
control problems, which is efficiently approximated by means of deep neural
networks and supervised learning, enabling the fast simulation of
high-dimensional, large-scale ensembles of controlled particles. Numerical
experiments demonstrate the effectiveness of the proposed approach in the
control of consensus and attraction-repulsion dynamics.
