initial conditions, we have defined an averaged relative $L_2$ error between the true and predicted trajectories $$\begin{aligned}
\label{error}
e(t)=\left[\frac{||X_{true}(t)-X_{pred}(t)||}{\left\langle||X_{true}(t)||\right\rangle}\right].\end{aligned}$$ Here $\left[ \cdot \right]$ and $\langle \cdot \rangle$ indicate, respectively, averaging over $100$ initial conditions and over $2000 \Delta t$. To be clear, $X_{true}(t)$ refers to the data at time $t$ obtained from the numerical solution while $X_{pred}(t)$ refers to the predicted value at $t$ using one of the deep learning methods. Figure \[esn\_time\](C) compares $e(t)$ for the three methods. It is clear that RC-ESN significantly outperforms ANN and RNN-LSTM for short-term prediction in this multi-scale chaotic testbed. Figure \[esn\_contour\] shows an example of the spatio-temporal evolution of $X_{pred}$ (from RC-ESN), $X_{true}$, and their difference, which further demonstrates the capabilities of RC-ESN for short-term spatio-temporal prediction.

![Performance of RC-ESN for short-term spatio-temporal prediction. We remind the readers that only the slow/large-scale variable $X$ has been used during training/testing, and the fast/small-scale variables $Y$ and $Z$ have not been used at any point during training or testing. RC-ESN has substantial forecast skills, providing accurate predictions up to $\approx 2$ MTU, which is around $400\Delta t$ or $9$ Lyapunov timescales. $N=500000$, algorithm $T_2$, $D=5000$, and $\rho=0.1$ are used. []{data-label="esn_contour"}](Figure2_MTUsmall.pdf){width="\textwidth"}

Short-term prediction: Scaling of 