  Modeling the complex interactions of systems of particles or agents is a
fundamental scientific and mathematical problem that is studied in diverse
fields, ranging from physics and biology, to economics and machine learning. In
this work, we describe a very general second-order, heterogeneous,
multivariable, interacting agent model, with an environment, that encompasses a
wide variety of known systems. We describe an inference framework that uses
nonparametric regression and approximation theory based techniques to
efficiently derive estimators of the interaction kernels which drive these
dynamical systems. We develop a complete learning theory which establishes
strong consistency and optimal nonparametric min-max rates of convergence for
the estimators, as well as provably accurate predicted trajectories. The
estimators exploit the structure of the equations in order to overcome the
curse of dimensionality and we describe a fundamental coercivity condition on
the inverse problem which ensures that the kernels can be learned and relates
to the minimal singular value of the learning matrix. The numerical algorithm
presented to build the estimators is parallelizable, performs well on
high-dimensional problems, and is demonstrated on complex dynamical systems.
