Recurrent neural networks and Koopman-based frameworks for temporal predictions in a low-order model of turbulence

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.

Automated summary

The study evaluated the capabilities of recurrent neural networks and Koopman-based frameworks in predicting the temporal dynamics of the low-order model of near-wall turbulence. The results showed that properly trained LSTM networks could achieve excellent reproductions of long-term statistics and dynamic behavior of the chaotic system, while the KNF framework offered the same level of accuracy in statistics at a lower computational cost.