$ X^{P_n} $ are polynomials of degree $n$. This includes cross terms of the single states. For the case of quadratic polynomials, this yields to Eq. . $$\delimitershortfall=0pt
\begin{footnotesize}
    X^{P_2} \!= \!
\begin{bmatrix}
x_1^2(t_1) \!&\! x_1(t_1)x_2(t_1) \!& \!\dots\! &\! x_2^2(t_1) &\! x_2(t_1)x_3(t_1) & \!\dots\! &\! x_n^2(t_1) \\
x_1^2(t_2) \!&\! x_1(t_2)x_2(t_2) \!& \!\dots\! &\! x_2^2(t_2) &\! x_2(t_2)x_3(t_2) & \!\dots\! &\! x_n^2(t_2) \\
\vdots     \!&\! \vdots           \!& \!\ddots\!&\! \vdots     &\!  \vdots        & \!\ddots\!&\!\vdots     \\
x_1^2(t_m) \!&\! x_1(t_m)x_2(t_m) \!& \!\dots\! &\! x_2^2(t_m) &\! x_2(t_m)x_3(t_m) & \!\dots\! &\! x_n^2(t_m)
    \end{bmatrix}\label{eq:polynomialLib}
\end{footnotesize}$$

So, the dynamic can be expressed by $\dot{X} = \Theta(X) \Xi$, where $\Theta(X)$ is the evaluated library and $\Xi = \begin{bmatrix}
\xi_1 & \xi_2 & \dots & \xi_n
\end{bmatrix}$ is the coefficient matrix. The coefficients can be determined using sparse regression techniques, e.g. sequential least-squares regression, as has been done in [@BruntonSINDy.2016].

For time-discrete systems, the equation $$\label{eq:sindyDiscrete}
X_{k+1} = \Theta(X_k) \Xi.$$ can be solved accordingly. SINDy allows for easy identification of few dominant functions out of a basis library. Choosing an appropriate library, which contains sufficient functions to describe the dynamics, is essential for a successful identification.

### Criteria for model quality.

There are several methods to compare different models and to 