Control of high-dimensional collective dynamics by deep neural feedback laws and kinetic modelling

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.