Learning Interaction Kernels for Agent Systems on Riemannian Manifolds
This addresses the challenge of modeling complex phenomena like opinion dynamics and predator-swarm systems on curved spaces, which is incremental as it extends existing kernel learning methods to Riemannian manifolds.
The paper tackles the problem of learning interaction kernels for agent systems constrained to evolve on Riemannian manifolds from trajectory data, showing that their estimators converge at a rate independent of the state space dimension and deriving error bounds for trajectory estimation on the manifold.
Interacting agent and particle systems are extensively used to model complex phenomena in science and engineering. We consider the problem of learning interaction kernels in these dynamical systems constrained to evolve on Riemannian manifolds from given trajectory data. The models we consider 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. We show that our estimators converge at a rate that is independent of the dimension of the state space, and derive bounds on the trajectory estimation error, on the manifold, between the observed and estimated dynamics. We demonstrate the performance of our estimator on two classical first order interacting systems: Opinion Dynamics and a Predator-Swarm system, with each system constrained on two prototypical manifolds, the $2$-dimensional sphere and the Poincaré disk model of hyperbolic space.