the evolution of $X(t)$.

Inspired by several recent studies (that are discussed below), we have focused on evaluating the performance of three deep learning techniques in accomplishing (1) and (2). These data-driven methods are

-   RC-ESN: Echo state network (ESN), a specialized type of recurrent neural network (RNN), which belongs to the family of reservoir computing (RC),

-   ANN: A deep feed-forward artificial neural network,

-   RNN-LSTM: An RNN with long short-term memory (LSTM).

We have focused on these three methods because they have either shown promising performance in past studies (RC-ESN and ANN), or they are considered state of the art in learning from sequential data (RNN-LSTM). There is a growing number of studies focused on using deep learning techniques for data-driven modeling of chaotic and turbulent systems, for example to improve weather/climate modeling and prediction. Some of these studies have been referenced above. Below, we briefly describe three sets of studies with closest relevance to our objective and approach. Pathak and co-workers [@pathak2018model; @pathak2017using; @lu2017reservoir] have recently shown promising results with RC-ESN for predicting short-term spatio-temporal evolution (item 1) and in replicating attractors for the Lorenz 63 and [Kuramoto-Sivashinsky]{} equations. The objective of our paper (items 1-2) and 