Learning the Koopman Eigendecomposition: A Diffeomorphic Approach

TL;DR

This paper introduces a diffeomorphic, data-driven method for learning Koopman eigenfunctions of nonlinear systems, enabling stable linear predictions and universal approximation with guarantees of asymptotic stability.

Contribution

It presents the first approach to connect operator, system, and learning theories using a diffeomorphic framework for Koopman eigenfunction learning.

Findings

Method guarantees asymptotic stability.

Achieves universal approximation of Koopman eigenfunctions.

Validated through simulation examples.

Abstract

We present a novel data-driven approach for learning linear representations of a class of stable nonlinear systems using Koopman eigenfunctions. By learning the conjugacy map between a nonlinear system and its Jacobian linearization through a Normalizing Flow one can guarantee the learned function is a diffeomorphism. Using this diffeomorphism, we construct eigenfunctions of the nonlinear system via the spectral equivalence of conjugate systems - allowing the construction of linear predictors for nonlinear systems. The universality of the diffeomorphism learner leads to the universal approximation of the nonlinear system's Koopman eigenfunctions. The developed method is also safe as it guarantees the model is asymptotically stable regardless of the representation accuracy. To our best knowledge, this is the first work to close the gap between the operator, system and learning theories. The efficacy of our approach is shown through simulation examples.