prediction horizon of RC-ESN. (C) Relative $L_2$ error averaged over $100$ randomly chosen initial conditions $e(t)$ (see Eq. (\[error\])). The dashed horizontal line marks $e=0.3$. Time is in MTU, where $1$ MTU $=$ $200 \Delta t \approx 4.5/\lambda_{max}$. Training size is $N=500000$.[]{data-label="esn_time"}](fig2_time_series_allmethods_new.eps){width="\textwidth"}

As seen in Figure \[esn\_time\](A), RC-ESN accurately predicts the time series for over $2.3$ MTU, which is equivalent to $460\Delta t$ and over $10.35$ Lyapunov timescales. Closer examination shows that the RC-RSN prediction follows the true trajectory well even up to $\approx 4$ MTU. The RNN-LSTM has the next best prediction performance (up to around $0.9$ MTU or $180 \Delta t$). The prediction from ANN is for around $0.6$ MTU or $120 \Delta t$. For the example in Figure \[esn\_time\](B), all methods have shorter prediction horizons, but RC-ESN still has the best performance (accurate prediction up to $\approx 0.7$ MTU), followed by ANN and RNN-LSTM with similar prediction accuracies ($\approx 0.3$ MTU).

Searching through all $100$ initial conditions, the best prediction with RNN-LSTM is up to $\approx 1.7$ MTU (equivalent to $340\Delta t$), and the best prediction with ANN is up to $\approx 1.2$ MTU (equivalent to $240\Delta t$),

To compare the results over all $100$ randomly chosen 