Dynamical systems via Koopman operators

Composition operators, Dynamic Extended Mode decompositions…

October 13, 2020 — April 9, 2021

dynamical systems
Hilbert space
kernel tricks
signal processing
time series
Figure 1

NB: Koopman B.O. Koopman (Koopman 1931) not S.J. Koopman, who also works in dynamical systems.

I do not know how Koopman operators work, especially in the learning setting, but maybe this fragment of abstract will do for now (Budišić, Mohr, and Mezić 2012):

A majority of methods from dynamical system analysis, especially those in applied settings, rely on Poincaré’s geometric picture that focuses on “dynamics of states.” While this picture has fueled our field for a century, it has shown difficulties in handling high-dimensional, ill-described, and uncertain systems, which are more and more common in engineered systems design and analysis of “big data” measurements. This overview article presents an alternative framework for dynamical systems, based on the “dynamics of observables” picture. The central object is the Koopman operator: an infinite-dimensional, linear operator that is nonetheless capable of capturing the full nonlinear dynamics.

There is a brief literature review in Klus et al. (2020).

The Chaos book folks describe it so:

the Koopman operator action on a state space function \(a(x)\) is to replace it by its downstream value time \(t\) later, \(a(x) \rightarrow a(x(t)),\) evaluated at the trajectory point \(x(t)\)

\[ \left[\mathcal{K}^{t} a\right](x)=a\left(f^{t}(x)\right)=\int_{\mathcal{M}} d y \mathcal{K}^{t}(x, y) a(y), \] \(\mathcal{K}^{t}(x, y)=\delta\left(y-f^{t}(x)\right)\)

1 Recursive estimation

See recursive identification for generic theory of learning under the distribution shift induced by a moving parameter vector.

2 Incoming

3 References

Arbabi. 2020. Introduction to Koopman Operator Theory of Dynamical Systems.”
Brunton, Steven L. 2019. Notes on Koopman Operator Theory.”
Brunton, Steven L., Brunton, Proctor, et al. 2016. Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control.” PLOS ONE.
Brunton, Steven L., Budišić, Kaiser, et al. 2022. Modern Koopman Theory for Dynamical Systems.” SIAM Review.
Brunton, Steven L., and Kutz. 2019. Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control.
Brunton, Steven L., Proctor, and Kutz. 2016. Discovering Governing Equations from Data by Sparse Identification of Nonlinear Dynamical Systems.” Proceedings of the National Academy of Sciences.
Budišić, Mohr, and Mezić. 2012. Applied Koopmanism.” Chaos: An Interdisciplinary Journal of Nonlinear Science.
Cvitanović, Artuso, Mainieri, et al. 2016. Koopman Modes.” In Chaos: Classical and Quantum.
Gilpin. 2023. Model Scale Versus Domain Knowledge in Statistical Forecasting of Chaotic Systems.” Physical Review Research.
Ishikawa, Fujii, Ikeda, et al. 2018. Metric on Nonlinear Dynamical Systems with Perron-Frobenius Operators.” arXiv:1805.12324 [Cs, Math, Stat].
Klus, Koltai, and Schütte. 2016. On the Numerical Approximation of the Perron-Frobenius and Koopman Operator.” Journal of Computational Dynamics.
Klus, Nüske, Peitz, et al. 2020. Data-Driven Approximation of the Koopman Generator: Model Reduction, System Identification, and Control.” Physica D: Nonlinear Phenomena.
Klus, Schuster, and Muandet. 2020. Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces.” Journal of Nonlinear Science.
Koopman. 1931. Hamiltonian Systems and Transformation in Hilbert Space.” Proceedings of the National Academy of Sciences.
Kutz, Brunton, Brunton, et al. 2016. Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems.
Li, Dietrich, Bollt, et al. 2017. Extended Dynamic Mode Decomposition with Dictionary Learning: A Data-Driven Adaptive Spectral Decomposition of the Koopman Operator.” Chaos: An Interdisciplinary Journal of Nonlinear Science.
Lin, Tian, Livescu, et al. 2021. Data-Driven Learning for the Mori-Zwanzig Formalism: A Generalization of the Koopman Learning Framework.” arXiv:2101.05873 [Cond-Mat].
Lusch, Kutz, and Brunton. 2018. Deep Learning for Universal Linear Embeddings of Nonlinear Dynamics.” Nature Communications.
Mauroy, and Goncalves. 2020. Koopman-Based Lifting Techniques for Nonlinear Systems Identification.” IEEE Transactions on Automatic Control.
Morrill, Kidger, Salvi, et al. 2020. “Neural CDEs for Long Time Series via the Log-ODE Method.” In.
Schmid. 2010. Dynamic Mode Decomposition of Numerical and Experimental Data.” Journal of Fluid Mechanics.
Schwantes, and Pande. 2015. Modeling Molecular Kinetics with tICA and the Kernel Trick.” Journal of Chemical Theory and Computation.
Tu, Rowley, Luchtenburg, et al. 2014. On Dynamic Mode Decomposition: Theory and Applications.” Journal of Computational Dynamics.
Williams, Kevrekidis, and Rowley. 2015. A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition.” Journal of Nonlinear Science.
Williams, Rowley, and Kevrekidis. 2015. A Kernel-Based Method for Data-Driven Koopman Spectral Analysis.” Journal of Computational Dynamics.