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Dynamical systems across many disciplines are modeled as interactingparticles or agents, with interaction rules that depend on a very small numberof variables (e.g. pairwise distances, pairwise differences of phases, etc...),functions of the state of pairs of agents. Yet, these interaction rules cangenerate self-organized dynamics, with complex emergent behaviors (clustering,flocking, swarming, etc.). We propose a learning technique that, givenobservations of states and velocities along trajectories of the agents, yieldsboth the variables upon which the interaction kernel depends and theinteraction kernel itself, in a nonparametric fashion. This yields an effectivedimension reduction which avoids the curse of dimensionality from thehigh-dimensional observation data (states and velocities of all the agents). Wedemonstrate the learning capability of our method to a variety of first-orderinteracting systems.

Learning Interaction Variables and Kernels from Observations of Agent-Based Systems