# Data-Driven Model Reduction and Transfer Operator Approximation

## Abstract

In this review paper, we will present different data-driven dimension reduction techniques for dynamical systems that are based on transfer operator theory as well as methods to approximate transfer operators and their eigenvalues, eigenfunctions, and eigenmodes. The goal is to point out similarities and differences between methods developed independently by the dynamical systems, fluid dynamics, and molecular dynamics communities such as time-lagged independent component analysis, dynamic mode decomposition, and their respective generalizations. As a result, extensions and best practices developed for one particular method can be carried over to other related methods.

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## Notes

1. A general time-homogeneous Itô stochastic differential equation is given by $$\mathrm {d}\mathbf {X}_{t}=-\alpha (\mathbf {X}_{t})\,\mathbf {X}_{t}\,\mathrm {d}t+\sigma (\mathbf {X}_{t})\,\mathrm {d}\mathbf {W}_{t}$$, where $$\alpha :\mathbb {R}^{d}\rightarrow \mathbb {R}^{d}$$ and $$\sigma :\mathbb {R}^{d}\rightarrow \mathbb {R}^{d\times d}$$ are coefficient functions, and $$\{\mathbf {W}_{t}\}_{t\ge 0}$$ is a d-dimensional standard Wiener process.

2. We call a stochastic process $$\{\mathbf {X}_{t}\}_{t\ge 0}$$ time-homogeneous, or autonomous, if it holds for every $$t\ge s\ge 0$$ that the distribution of $$\mathbf {{X}}_{t}$$ conditional to $$\mathbf {X}_{s}=x$$ only depends on x and $$(t-s)$$. It is the stochastic analogue of the flow of an autonomous (time-independent) ordinary differential equation.

3. For a measure-theoretic discussion of this construction, please refer to Klus et al. (2016). For our purposes, it is sufficient to equip $$\mathbb {X}$$ with the standard Lebesgue measure. In particular, if not stated otherwise, measurability of a set $$\mathbb {A\subset X}$$ is meant with respect to the Borel $$\sigma$$-algebra.

4. These conditions are called interchangeably absolute continuity,$$\mu$$-compatibility, or null preservingness.

5. Algorithm for Multiple Unknown Signals Extraction.

6. The easiest way to accomplish this is by adding the observables $$x_{i}$$, $$i=1,\dots ,d$$, to the set of basis functions.

7. A process $$\{\mathbf {X}_{t}\}_{t\ge 0}$$ is called Feller-continuous if the mapping $$x\mapsto \mathbb {E}[g(\mathbf {X}_{t})\vert \mathbf {X}_{0}=x]$$ is continuous for any fixed continuous function g. This implies, that the Koopman operator of a Feller-continuous process has a well-defined restriction from $$L^{\infty }(\mathbb {X})$$ to the set of continuous functions. Any stochastic process generated by an Itô stochastic differential equation with Lipschitz-continuous coefficients is Feller-continuous (Øksendal 2003, Lemma 8.1.4).

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## Acknowledgements

This research has been partially funded by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 “Scaling Cascades in Complex Systems,” Project A04 “Efficient calculation of slow and stationary scales in molecular dynamics” and Project B03 “Multilevel coarse graining of multi-scale problems”, and by the Einstein Foundation Berlin (Einstein Center ECMath). Furthermore, we would like to thank the reviewers for their helpful comments and suggestions.

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Correspondence to Stefan Klus.

Communicated by Clarence W. Rowley.

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Klus, S., Nüske, F., Koltai, P. et al. Data-Driven Model Reduction and Transfer Operator Approximation. J Nonlinear Sci 28, 985–1010 (2018). https://doi.org/10.1007/s00332-017-9437-7

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• DOI: https://doi.org/10.1007/s00332-017-9437-7

### Keywords

• Koopman operator
• Perron-Frobenius operator
• Model reduction
• Data-driven methods

• 37M10
• 37M25
• 37L65
• 34L16