Maximum-likelihood reprojections for reliable Koopman-based predictions and bifurcation analysis of parametric dynamical systems

TL;DR

This paper introduces a maximum-likelihood reprojection method for Koopman-based models that improves prediction reliability and enables bifurcation analysis in parametric dynamical systems.

Contribution

It develops a reprojection approach based on closest-point projections to ensure consistency with the Koopman manifold, extending to parametric systems for bifurcation analysis.

Findings

Reprojection improves the accuracy of Koopman-based predictions.

The method is applicable to parametric systems for bifurcation analysis.

Reprojection aligns finite-data approximations with the nonlinear manifold.

Abstract

Koopman-based methods leverage a nonlinear lifting to enable linear regression techniques. Consequently, data generation, learning and prediction is performed through the lens of this lifting, giving rise to a nonlinear manifold that is invariant under the Koopman operator. In data-driven approximation such as Extended Dynamic Mode Decomposition, this invariance is typically lost due to the presence of (finite-data) approximation errors. In this work, we show that reprojections are crucial for reliable predictions. We provide an approach via closest-point projections that ensure consistency with this nonlinear manifold, which is strongly related to a Riemannian metric and maximum likelihood estimates. While these results are already novel for autonomous systems, we present our approach for parametric systems, providing the basis for data-driven bifurcation analysis and control applications.