Learning Feature Maps of the Koopman Operator: A Subspace Viewpoint

Lian, Yingzhao; Jones, Colin N.

doi:10.1109/cdc40024.2019.9029189

Cited by 10 publications

(7 citation statements)

References 32 publications

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“…Thus, for K t f with eigenvalues e λ1t and e λ2t , φ 1 φ 2 is also an eigenfunction of K t f with eigenvalue e (λ1+λ2)t . Definition 3.1 ( [17]): Consider {E p } p∈N to be the eigenpair group of the Koopman operator semigroup K t A of a system ẏ = Ay with its minimal generator G E : (10) where m i ∈ N 0 . Then, the elements of G E are principle eigenvalues with corresponding eigenfunctions of K t A in {E p } p∈N .…”

Section: Modeling Via Equivalence Relationsmentioning

confidence: 99%

“…Then, the elements of G E are principle eigenvalues with corresponding eigenfunctions of K t A in {E p } p∈N . Less formally, principle eigenpairs form the minimal set used to construct arbitrarily many other eigenpairs (10).…”

Section: Modeling Via Equivalence Relationsmentioning

confidence: 99%

“…-commonly leading to only locally accurate models and no practicable learning guarantees. Other approaches leverage the expressive power of neural networks or kernel methods to learn a suitable library of functions [9], [10] but often lack theoretical justification.…”

Section: Introductionmentioning

confidence: 99%

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Learning the Koopman Eigendecomposition: A Diffeomorphic Approach

Bevanda¹,

Kirmayr²,

Sosnowski³

et al. 2021

Preprint

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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.

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Section: Modeling Via Equivalence Relationsmentioning

confidence: 99%

Section: Modeling Via Equivalence Relationsmentioning

confidence: 99%

See 1 more Smart Citation

Learning the Koopman Eigendecomposition: A Diffeomorphic Approach

Bevanda¹,

Kirmayr²,

Sosnowski³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…However, these aforementioned methods mainly consider onestep forward prediction either due to the formulation of the learning problem or due to numerical stability, which results in a relatively inaccurate long-term prediction. A link between the Koopman operator and the subspace identification is observed in [33], which enables a learning scheme of long-term prediction. However, this method still suffers from the lack of scalability and the numerical limitation of the subspace identification [34].…”

Section: Introductionmentioning

confidence: 99%

Koopman based data-driven predictive control

Lian,

Wang,

Jones

2021

Preprint

Self Cite

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Sparked by the Willems' fundamental lemma, a class of data-driven control methods has been developed for LTI system. At the same time, the Koopman operator theory attempts cast a nonliner control problem into a standard linear one albeit infinite dimensional. Motivated by these two ideas, a data-driven control scheme for nonlinear systems is proposed in this work. The proposed scheme is compatible with most differential regressor enabling an offline learning. In particular, the model uncertainty is considered, enabling a novel data-driven simulation framework based on Wasserstein distance. Numerical experiments are performed with Bayesian neural networks to show the effectiveness of both the proposed control and simulation scheme.

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“…However, presupposing a suitable basis of functions is a very strong assumption for linear time-invariant prediction -leading to only locally accurate models. Other approaches learn the features simultaneously (Li et al, 2017) or in a decoupled manner (Lian and Jones, 2019) leveraging the expressive power of neural networks or kernel methods, but often lack theoretical justification. Instead of arbitrary feature maps, Korda and Mezić (2020) learn the eigenfunctions of the operator for linear prediction -which is a promising proposition.…”

Section: Introductionmentioning

confidence: 99%

Diffeomorphically Learning Stable Koopman Operators

Bevanda¹,

Beier²,

Sebastian³

et al. 2021

Preprint

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We propose a novel framework for constructing linear time-invariant (LTI) models for datadriven 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-ofthe-art method on the well-known LASA handwriting dataset.

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Learning Feature Maps of the Koopman Operator: A Subspace Viewpoint

Cited by 10 publications

References 32 publications

Learning the Koopman Eigendecomposition: A Diffeomorphic Approach

Learning the Koopman Eigendecomposition: A Diffeomorphic Approach

Koopman based data-driven predictive control

Diffeomorphically Learning Stable Koopman Operators

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