row of the matrix $\mathbf{r}$ is a snapshot $r(t)$ of size $D$). For example, following Pathank *et al*.  [@pathak2018model], we compute $\tilde{\mathbf{r}}$ as having the same even columns as ${\mathbf{r}}$ while its odd columns are the square of the odd columns of ${\mathbf{r}}$ (algorithm $T_1$ hereafter). As shown in Appendix \[sec:appnT\], we have found that a nonlinear transformation (between ${\mathbf{r}}$ and $\tilde{\mathbf{r}}$) is essential for skillful predictions, while several other transformation algorithms yield similar results as $T_1$. The choices of these transformations ($T_2$ and $T_3$, see Appendix \[sec:appnT\]), although not based on a rigorous mathematical analysis, are inspired from the nature of the quadratic nonlinearity that is present in Eqs. (\[eq:L1\])-(\[eq:L3\]).

The prediction process is governed by: $$\begin{aligned}
\label{esn_eqn_test}
v(t+\Delta t)=\mathbf{W_{out}}\tilde{r}(t+\Delta t), \label{eq:v} \\
X(t+\Delta t)=v(t+\Delta t). \label{eq:tdt}\end{aligned}$$ $\tilde{r}(t+\Delta t)$ in Eq. (\[eq:v\]) is computed by applying one of the $T_1$, $T_2$ or $T_3$ algorithms on $r(t+\Delta t)$, which itself is calculated via Eq. (\[eq:r\]) from $X(t)$ that is either known from initial condition or has been previously predicted.

See [@jaeger2004harnessing; @Jaeger2007; @lukovsevivcius2009reservoir; @gauthier2018reservoir] and references therein for further discussions on RC-ESNs, and [@lu2017reservoir; @pathak2017using; @mcdermott2017ensemble; @pathak2018model; @pathak2018hybrid; @zimmermann2018observing; @lu2018attractor; @mcdermott2019deep; @lim2019predicting] for examples of recent applications to dynamical systems.

