• Corpus ID: 239009622

Learning the Koopman Eigendecomposition: A Diffeomorphic Approach

@article{Bevanda2021LearningTK,
  title={Learning the Koopman Eigendecomposition: A Diffeomorphic Approach},
  author={Petar Bevanda and Johannes Kirmayr and Stefan Sosnowski and Sandra Hirche},
  journal={ArXiv},
  year={2021},
  volume={abs/2110.07786}
}
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… 

Figures and Tables from this paper

References

SHOWING 1-10 OF 25 REFERENCES
Learning Feature Maps of the Koopman Operator: A Subspace Viewpoint
TLDR
A new data-driven framework for learning feature maps of the Koopman operator by introducing a novel separation method that provides a flexible interface between diverse machine learning algorithms and well-developed linear subspace identification methods.
Optimal Construction of Koopman Eigenfunctions for Prediction and Control
TLDR
This article presents a novel data-driven framework for constructing eigenfunctions of the Koopman operator geared toward prediction and control, and is extended to construct generalized eigenFunctions that also give rise Koop man invariant subspaces and hence can be used for linear prediction.
Extended Dynamic Mode Decomposition with Learned Koopman Eigenfunctions for Prediction and Control
TLDR
This method is the first to utilize Koopman eigenfunctions as lifting functions for EDMD-based methods and is able to significantly improve the controller performance while relying on linear control methods to do nonlinear control.
A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
TLDR
This approach is an extension of dynamic mode decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes, and if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation.
Linearly-Recurrent Autoencoder Networks for Learning Dynamics
TLDR
A new neural network architecture combining an autoencoder with linear recurrent dynamics in the encoded state is used to learn a low-dimensional and highly informative Koopman-invariant subspace of observables.
Linearization in the large of nonlinear systems and Koopman operator spectrum
Abstract According to the Hartman–Grobman Theorem, a nonlinear system can be linearized in a neighborhood of a hyperbolic stationary point. Here, we extend this linearization around stable (unstable)
On Matching, and Even Rectifying, Dynamical Systems through Koopman Operator Eigenfunctions
TLDR
It is argued that the use of the Koopman operator and its spectrum is particularly well suited for this endeavor, both in theory, but also especially in view of recent data-driven algorithm developments.
Data-driven approximation of the Koopman generator: Model reduction, system identification, and control
We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This
Koopman Operator Dynamical Models: Learning, Analysis and Control
TLDR
This work reviews data-driven representations for Koopman operator dynamical models, categorizing various existing methodologies and highlighting their differences, and provides concise insight into the paradigm’s relation to system-theoretic notions and analyzes the prospect of using the paradigm for modeling control systems.
Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems
TLDR
This doctoral dissertation focuses on the Koopman, or composition, operator that determines how a function on the state space evolves as the state trajectories evolve, and develops the Generalized Laplace Analysis (GLA) for both spectral operators of scalar type and non spectral operators.
...
1
2
3
...