KoopmanizingFlows: Diffeomorphically Learning Stable Koopman Operators

• URL: http://arxiv.org/abs/2112.04085v1
• Date: Wed, 8 Dec 2021 02:40:40 GMT
• Title: KoopmanizingFlows: Diffeomorphically Learning Stable Koopman Operators
• Authors: Petar Bevanda, Max Beier, Sebastian Kerz, Armin Lederer, Stefan Sosnowski and Sandra Hirche
• Abstract summary: We propose a novel framework for constructing linear time-invariant (LTI) models for a class of stable nonlinear dynamics. We learn the Koopman operator features without assuming a predefined library of functions or knowing the spectrum. We demonstrate the superior efficacy of the proposed method in comparison to a state-of-the-art method on the well-known LASA handwriting dataset.
• Score: 7.447933533434023
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose a novel framework for constructing linear time-invariant (LTI) models for data-driven representations of the Koopman operator for a class of stable nonlinear dynamics. The Koopman operator (generator) lifts a finite-dimensional nonlinear system to a possibly infinite-dimensional linear feature space. To utilize it for modeling, one needs to discover finite-dimensional representations of the Koopman operator. Learning suitable features is challenging, as one needs to learn LTI features that are both Koopman-invariant (evolve linearly under the dynamics) as well as relevant (spanning the original state) - a generally unsupervised learning task. For a theoretically well-founded solution to this problem, we propose learning Koopman-invariant coordinates by composing a diffeomorphic learner with a lifted aggregate system of a latent linear model. Using an unconstrained parameterization of stable matrices along with the aforementioned feature construction, we learn the Koopman operator features without assuming a predefined library of functions or knowing the spectrum, while ensuring stability regardless of the operator approximation accuracy. We demonstrate the superior efficacy of the proposed method in comparison to a state-of-the-art method on the well-known LASA handwriting dataset.

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