  Systems of interacting particles or agents have wide applications in many
disciplines such as Physics, Chemistry, Biology and Economics. These systems
are governed by interaction laws, which are often unknown: estimating them from
observation data is a fundamental task that can provide meaningful insights and
accurate predictions of the behaviour of the agents. In this paper, we consider
the inverse problem of learning interaction laws given data from multiple
trajectories, in a nonparametric fashion, when the interaction kernels depend
on pairwise distances. We establish a condition for learnability of interaction
kernels, and construct estimators that are guaranteed to converge in a suitable
$L^2$ space, at the optimal min-max rate for 1-dimensional nonparametric
regression. We propose an efficient learning algorithm based on least squares,
which can be implemented in parallel for multiple trajectories and is therefore
well-suited for the high dimensional, big data regime. Numerical simulations on
a variety examples, including opinion dynamics, predator-swarm dynamics and
heterogeneous particle dynamics, suggest that the learnability condition is
satisfied in models used in practice, and the rate of convergence of our
estimator is consistent with the theory. These simulations also suggest that
our estimators are robust to noise in the observations, and produce accurate
predictions of dynamics in relative large time intervals, even when they are
learned from data collected in short time intervals.
