  Dynamical systems across many disciplines are modeled as interacting
particles or agents, with interaction rules that depend on a very small number
of variables (e.g. pairwise distances, pairwise differences of phases, etc...),
functions of the state of pairs of agents. Yet, these interaction rules can
generate self-organized dynamics, with complex emergent behaviors (clustering,
flocking, swarming, etc.). We propose a learning technique that, given
observations of states and velocities along trajectories of the agents, yields
both the variables upon which the interaction kernel depends and the
interaction kernel itself, in a nonparametric fashion. This yields an effective
dimension reduction which avoids the curse of dimensionality from the
high-dimensional observation data (states and velocities of all the agents). We
demonstrate the learning capability of our method to a variety of first-order
interacting systems.
