Learning Interaction Kernels for Agent Systems on Riemannian Manifolds

Abstract

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.

References in corpus (11)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• Deep learning of dynamics and signal-noise decomposition with time-stepping constraints
• Sparse Dynamics for Partial Differential Equations
• Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems
• Nonparametric inference of interaction laws in systems of agents from trajectory data
• Learning interaction kernels in heterogeneous systems of agents from multiple trajectories
• Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets
• Discovery of Dynamics Using Linear Multistep Methods
• On the identifiability of interaction functions in systems of interacting particles
• Multi-Agent Systems based on Contextual Defeasible Logic considering Focus
• Inference of dynamic systems from noisy and sparse data via manifold-constrained Gaussian processes