RC-ESN and ANN performance with training size $N$ {#define_E}
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How the performance of deep learning techniques scales with the size of the training set is of significant practical importance as the amount of available data is often limited in many problems. Given that currently there is no theoretical understanding of such scaling for these deep learning techniques, we have empirically examined how the quality of short-term predictions scales with $N$. We have conducted the scaling analysis for $N=10^4$ to $N=2 \times 10^6$ for the three methods. Two metrics for the quality of prediction are used: the prediction horizon, defined as when the averaged $L_2$ error $e(t)$ reaches $0.3$, and the prediction error $E$, defined as the average of $e(t)$ between $0-0.5$ MTU: $$\begin{aligned}
\label{eq:E}
E=\frac{1}{100\Delta t}\sum_{i=0}^{i=100}e(i\Delta t).\end{aligned}$$

Figure \[esn\_scaling\](A) shows that for all methods, the prediction horizon increases monotonically but nonlinearly as we increase $N$. The prediction horizons of RC-ESN and ANN appear to saturate after $N=10^6$, although the RC-ESN has a more complex step-like scaling curve that needs further examination in future studies. The prediction horizon of RC-ESN exceeds that of ANN by factors ranging from $3$ (for high $N$) to $9$ (for low $N$). In the case 