Learning Interaction Kernels and Emergent Behaviors for Second Order Interacting Agent Systems

Abstract

Modeling the complex interactions of systems of particles or agents is a fundamental problem across the sciences, from physics and biology, to economics and social sciences. In this work, we consider second-order, heterogeneous, multivariable models of interacting agents or particles, within simple environments. We describe a nonparametric inference framework to efficiently estimate the interaction kernels which drive these dynamical systems. We develop a complete learning theory which establishes strong consistency and optimal nonparametric min-max rates of convergence for the estimators, as well as provably accurate predicted trajectories. The estimators exploit the structure of the equations in order to overcome the curse of dimensionality; furthermore we describe a fundamental coercivity condition which ensures that the interaction kernels can be learned and relates to the minimal singular value of the learning matrix. The numerical algorithm presented to build the estimators is parallelizable, performs well on high-dimensional problems, and its performance is tested on a variety of complex dynamical systems.

We are often interested in collective dynamical systems exhibiting emergent behaviors with complicated interaction kernels, and with kernels which are parameterized by a single unknown parameter. We provide extensive numerical evidence that the estimators provide faithful approximations to these 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 much beyond. We demonstrate these features on prototypical systems displaying collective behaviors, ranging from opinion dynamics, flocking dynamics, self-propelling particle dynamics, to synchronized oscillator dynamics. We also consider the problem of learning interaction kernels in these dynamical systems constrained to evolve on Riemannian manifolds. The models are based on interaction kernels depending on pairwise Riemannian distances between agents, with agents interacting locally along the direction of the shortest geodesic connecting them.

Lastly, we build accurate and predictive models of the underlying mechanisms of celestial motion. By modeling the major Astronomical Bodies in the Solar system as pairwise interacting bodies, we generate extremely accurate dynamics can provide a unified explanation to the observation data, especially in terms of reproducing the perihelion precession of Mars, Mercury, and the Moon.

Primary Reader: Mauro Maggioni

Secondary Reader: Fei Lu