Recurrent neural networks and Koopman-based frameworks for temporal predictions in turbulence.

The prediction capabilities of recurrent neural networks and Koopman-based frameworks are assessed in 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 predictions of the turbulence statistics and the dynamic behavior of the flow 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 flow can lead to suboptimal predictions in terms of turbulence statistics. Thus, we propose a stopping criterion based on the computed statistics which effectively avoids overfitting to instantaneous predictions at the cost of deteriorated statistics. This suggests that a new loss function, including the averaged behavior of the flow as well as the instantaneous predictions, may lead to an improved generalization ability of the network.