lookback) [@kim1999nonlinear]. An extensive hyperparameter optimization (by trial and error) is performed to find the optimal value of $q$ for which the network has the largest prediction horizon (exploroing $q=1-22$, we found $q=3$ to yield the best performance). The RNN-LSTM predicts $X(t+\Delta t)$ from the previous $q$ time steps of $X(t)$. This is in contrast with RC-ESN, which only uses $X(t)$ and reservoir state $r(t)$ to predict $X(t+\Delta t)$, and ANN, which only uses $X(t)$ and no state to predict $X(t+\Delta t)$ (via predicting $\Delta X(t)$). The weights of the LSTM layers are determined during the training process (see \[sec:lstm\]). During testing, $X(t+\Delta t)$ is predicted using the past $q$ observables $\left[X(t-(q-1)\Delta t) \dots X(t-\Delta t), X(t)\right]$ that are either known from initial condition or have been previously predicted. We have found the best results with a stateless LSTM (which means that the state of the LSTM gets refreshed during the beginning of each batch during training, see \[sec:lstm\]) that outperforms a stateful LSTM (where the states gets carried over to the next batch during training).

The architecture of our RNN-LSTM is similar to the one used in Vlachas *et al*.  [@vlachas2018data]. There is no overfitting in the training phase 