of RNN-LSTM, the factor ranges from $1.2$ (for high $N$) to $2$ (for low $N$). Figure \[esn\_scaling\](B) shows that for all methods, the average error $E$ decreases as $N$ increases (as expected), most notably for ANN when $N$ is small.

![Comparison of the short-term prediction quality for RC-ESN (blue circles), RNN-LSTM (black stars), and ANN (red squared) as the size of the training set $N$ is changed from $N=10^4$ to $N=2 \times 10^6$. (A) Prediction horizon (when $e(t)$ reaches $0.3$). (B) Average error $E$ (see Eq. (\[eq:E\])). The hyperparameters in each method are optimized for each $N$.[]{data-label="esn_scaling"}](Figure3_scaling_new.eps){width="\textwidth"}

Overall, compared to both RNN-LSTM and ANN, the prediction horizon and accuracy of RC-ESN have a weaker dependence on the size of the training set, which is a significant advantage for RC-ESN when the dataset available for training is short, which is common in many practical problems.

Short-term prediction: Scaling of RC-ESN performance with reservoir size $D$
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Given the superior performance of RC-ESN for short-term prediction, here we focus on one concern with this method: the need for large reservoirs, which can be computationally demanding. This issue has been suggested as a potential disadvantage of ESNs versus LSTMs for training RNNs [@Jaeger2007]. 