Data-driven discovery of emergent behaviors in collective dynamics.

Particle- and agent-based systems are a ubiquitous modeling tool in many disciplines. We consider the fundamental problem of inferring the governing structure, i.e. interaction kernels, in a nonparametric fashion, from observations of agent-based dynamical systems. In particular, we are interested in collective dynamical systems exhibiting emergent behaviors with complicated interaction kernels, and for kernels which are parameterized by a single unknown parameter. This work extends the estimators introduced in Lu et al. (2019), 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, to a gravitational system. Our experiments also suggest that our estimated systems can display the same emergent behaviors as the observed systems, including those that occur at larger timescales than those in the training data. Finally, in the case of families of systems governed by a parametric family of interaction kernels, we introduce novel estimators that estimate the parametric 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/