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Particle- and agent-based systems are a ubiquitous modeling tool in manydisciplines. We consider the fundamental problem of inferring interactionkernels from observations of agent-based dynamical systems given observationsof trajectories, in particular for collective dynamical systems exhibitingemergent behaviors with complicated interaction kernels, in a nonparametricfashion, and for kernels which are parametrized by a single unknown parameter.We extend the estimators introduced in \cite{PNASLU}, which are based onsuitably regularized least squares estimators, to these larger classes ofsystems. We provide extensive numerical evidence that the estimators providefaithful approximations to the interaction kernels, and provide accuratepredictions for trajectories started at new initial conditions, both throughoutthe ``training'' time interval in which the observations were made, and oftenmuch beyond. We demonstrate these features on prototypical systems displayingcollective behaviors, ranging from opinion dynamics, flocking dynamics,self-propelling particle dynamics, synchronized oscillator dynamics, and agravitational system. Our experiments also suggest that our estimated systemscan display the same emergent behaviors of the observed systems, that occur atlarger timescales than those used in the training data. Finally, in the case offamilies of systems governed by a parameterized family of interaction kernels,we introduce novel estimators that estimate the parameterized family ofkernels, splitting it into a common interaction kernel and the action ofparameters. 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 prioriknowledge of either one, from observations of planetary motions in our solarsystem.

Data-driven Discovery of Emergent Behaviors in Collective Dynamics