Diffeomorphically Learning Stable Koopman Operators

TL;DR

This paper introduces Koopmanizing Flows, a novel continuous-time supervised learning framework that constructs stable, finite-dimensional linear predictors for nonlinear systems using diffeomorphic transformations, improving prediction accuracy and stability.

Contribution

It proposes a new method combining diffeomorphic transformations with Koopman operator theory to learn stable, linear predictors for nonlinear dynamics in a supervised manner.

Findings

Outperforms state-of-the-art methods on LASA handwriting benchmark

Ensures asymptotic stability through unconstrained Hurwitz matrix parameterization

Effectively learns linear predictors for complex nonlinear systems

Abstract

System representations inspired by the infinite-dimensional Koopman operator (generator) are increasingly considered for predictive modeling. Due to the operator's linearity, a range of nonlinear systems admit linear predictor representations - allowing for simplified prediction, analysis and control. However, finding meaningful finite-dimensional representations for prediction is difficult as it involves determining features that are both Koopman-invariant (evolve linearly under the dynamics) as well as relevant (spanning the original state) - a generally unsupervised problem. In this work, we present Koopmanizing Flows - a novel continuous-time framework for supervised learning of linear predictors for a class of nonlinear dynamics. In our model construction a latent diffeomorphically related linear system unfolds into a linear predictor through the composition with a monomial basis. The lifting, its linear dynamics and state reconstruction are learned simultaneously, while an unconstrained parameterization of Hurwitz matrices ensures asymptotic stability regardless of the operator approximation accuracy. The superior efficacy of Koopmanizing Flows is demonstrated in comparison to a state-of-the-art method on the well-known LASA handwriting benchmark.