  Particle- and agent-based systems are a ubiquitous modeling tool in many
disciplines. We consider the fundamental problem of inferring interaction
kernels from observations of agent-based dynamical systems given observations
of trajectories, in particular for collective dynamical systems exhibiting
emergent behaviors with complicated interaction kernels, in a nonparametric
fashion, and for kernels which are parametrized by a single unknown parameter.
We extend the estimators introduced in \cite{PNASLU}, which are based on
suitably regularized least squares estimators, to these larger classes of
systems. We provide extensive numerical evidence that the estimators provide
faithful approximations to the interaction kernels, and provide accurate
predictions for trajectories started at new initial conditions, both throughout
the ``training'' time interval in which the observations were made, and often
much beyond. We demonstrate these features on prototypical systems displaying
collective behaviors, ranging from opinion dynamics, flocking dynamics,
self-propelling particle dynamics, synchronized oscillator dynamics, and a
gravitational system. Our experiments also suggest that our estimated systems
can display the same emergent behaviors of the observed systems, that occur at
larger timescales than those used in the training data. Finally, in the case of
families of systems governed by a parameterized family of interaction kernels,
we introduce novel estimators that estimate the parameterized family of
kernels, splitting it into a common interaction kernel and the action of
parameters. We demonstrate this in the case of gravity, by learning both the
``common component'' $1/r^2$ and the dependency on mass, without any a priori
knowledge of either one, from observations of planetary motions in our solar
system.
