**4. Koopman theory and dynamic mode decomposition**

#### **4.1 Koopman operator for discrete dynamical systems**

Koopman theory is a powerful mathematical framework that re-expresses a general nonlinear discrete dynamical system as the knowledge of a linear (infinite dimensional) operator, the so-called Koopman operator or compositional operator. Today it is commonly used in machine learning and data-driven model-order reduction methodologies [4, 12]. Let us assume a discrete dynamical system in the form

$$\mathfrak{a}^{n+1} = F(\mathfrak{a}^n), \quad n \in \mathbb{N} \tag{15}$$

for a Lipschitz continuous mapping *F* from *<sup>d</sup>* to *<sup>d</sup>*. Let *g* be a function of a Banach space *<sup>X</sup>*, *<sup>g</sup>* : *<sup>d</sup>* ! . So we have *<sup>g</sup> <sup>a</sup><sup>n</sup>*þ<sup>1</sup> ð Þ¼ ð Þ *<sup>g</sup>* <sup>∘</sup> *<sup>F</sup> <sup>a</sup><sup>n</sup>* ð Þ. The Koopman operator related to *F* is defined as

$$
\mathbb{X}\mathfrak{g} = \mathfrak{g} \circ \mathbf{F} \quad \forall \mathfrak{g} \in \mathbf{X}.\tag{16}
$$

Then we have

$$\lg(\mathfrak{a}^{n+1}) = (\lhd \mathfrak{g})(\mathfrak{a}^{n}).\tag{17}$$

The knowledge of K includes the knowledge of *F*. Indeed, by taking the particular observables *gi* ð Þ¼ *a a* � *ei*, *i* ¼ 1, … , *d* where *e<sup>i</sup>* is the *i*-th vector of the canonical basis of *<sup>d</sup>*, we retrieve *<sup>a</sup><sup>n</sup>*þ<sup>1</sup> ð Þ*<sup>i</sup>* <sup>¼</sup> *Fi <sup>a</sup><sup>n</sup>* ð Þ, i.e. the *<sup>i</sup>*-th equation of (15). Of course, the linear Koopman operator acts on an infinite-dimensional functional space, so it is impossible to determine it exactly. However, one can search for an approximate Koopman operator K~ that acts on an approximate finite-dimensional space *X*~ ⊂*X*.

The concept of (nonlinear) observables is to have a sufficiently large set of independent nonlinear functions of the state vector and measurements of them in order to identify the mapping *F*. A natural question of interest is what are the best observables to choose. There is no absolute answer to this question, and the choice may depend on the underlying Physics. Without any a priori knowledge on the system of equations, one can use basis functions of a universal approximators like polynomials, Fourier or kernel-based functions for example.

## **4.2 Dynamic mode decomposition**

The simplest choice of observables is the linear functions *gi* ð Þ¼ *a a* � *ei*, *i* ¼ 1, … , *d*. It leads to the search of a finite-dimensional approximation *A* of T from the full state vector data. The matrix *A* can be searched as the solution of the least square minimization problem

$$\min\_{A \in \mathcal{M}\_d(\mathbb{R})} \quad \frac{1}{2} \|Y - AX\|\_F^2 \tag{18}$$

where *<sup>X</sup>* <sup>¼</sup> *<sup>a</sup>*0, *<sup>a</sup>*1, … , *<sup>a</sup><sup>N</sup>*�<sup>1</sup> � � and *<sup>Y</sup>* <sup>¼</sup> *<sup>a</sup>*1, *<sup>a</sup>*2, … , *<sup>a</sup><sup>N</sup>* � �. Assuming *<sup>N</sup>* <sup>≥</sup>*d*, the solution is given by *<sup>A</sup>* <sup>¼</sup> *YX<sup>T</sup> XX<sup>T</sup>* � ��<sup>1</sup> . The least square problem (18) can be eventually regularized for better conditioning by a Tykhonov regularization term [13, 14]. This practical approach of Koopman operator approximation is referred to as the dynamic mode decomposition (DMD) [4, 12]. This provides a linear dynamical model

$$\mathbf{a}^{n+1} = \mathbf{A}\mathbf{a}^{n} \tag{19}$$

starting from a given initial condition *<sup>a</sup>*0. The solution of (19) which is *<sup>a</sup><sup>n</sup>* <sup>¼</sup> *Ana*<sup>0</sup> is bounded for any initial condition *<sup>a</sup>*<sup>0</sup> as soon as *<sup>ρ</sup>*ð Þ *<sup>A</sup>* <sup>≤</sup>1.
