Propagation of Uncertainty with the Koopman Operator

TL;DR

This paper introduces a novel approach for propagating uncertainties in nonlinear systems using the Koopman Operator, enabling direct probability density function evolution through basis function projections.

Contribution

It presents a new method combining analytical and numerical Koopman Operator techniques for uncertainty propagation with a recursive least squares reduction algorithm.

Findings

Effective propagation of probability densities in nonlinear dynamics.

Compatibility with both Galerkin and EDMD Koopman approximations.

Recursivity achieved through least squares reduction.

Abstract

This paper proposes a new method to propagate uncertainties undergoing nonlinear dynamics using the Koopman Operator (KO). Probability density functions are propagated directly using the Koopman approximation of the solution flow of the system, where the dynamics have been projected on a well-defined set of basis functions. The prediction technique is derived following both the analytical (Galerkin) and numerical (EDMD) derivation of the KO, and a least square reduction algorithm assures the recursivity of the proposed methodology.