  The capabilities of recurrent neural networks and Koopman-based frameworks
are assessed in the prediction of temporal dynamics of the low-order model of
near-wall turbulence by Moehlis et al. (New J. Phys. 6, 56, 2004). Our results
show that it is possible to obtain excellent reproductions of the long-term
statistics and the dynamic behavior of the chaotic system with properly trained
long-short-term memory (LSTM) networks, leading to relative errors in the mean
and the fluctuations below $1\%$. Besides, a newly developed Koopman-based
framework, called Koopman with nonlinear forcing (KNF), leads to the same level
of accuracy in the statistics at a significantly lower computational expense.
Furthermore, the KNF framework outperforms the LSTM network when it comes to
short-term predictions. We also observe that using a loss function based only
on the instantaneous predictions of the chaotic system can lead to suboptimal
reproductions in terms of long-term statistics. Thus, we propose a
model-selection criterion based on the computed statistics which allows to
achieve excellent statistical reconstruction even on small datasets, with
minimal loss of accuracy in the instantaneous predictions.
