Data-driven Discovery of Emergent Behaviors in Collective Dynamics

Abstract: 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 thecommon 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.