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.

Moehlis等人在预测近壁湍流低阶模型的时间动态时评估了递归神经网络和基于Koopman框架的功能。（新泽西J.的PHY。6，56，2004）。我们的结果表明，通过训练有素的长期短期记忆（LSTM）网络，可以获得长期统计数据和混沌系统动态行为的出色再现，从而导致均值相对误差和下面的波动$\mathrm{1个}％$。此外，新开发的基于Koopman的框架，称为具有非线性强迫（KNF）的Koopman，可以显着降低计算开销，从而达到相同级别的统计精度。此外，就短期预测而言，KNF框架优于LSTM网络。我们还观察到，仅基于混沌系统的瞬时预测来使用损失函数，就长期统计而言可能导致次优的复制。因此，我们提出了一种基于计算统计量的模型选择标准，即使在较小的数据集上，该模型选择标准也可以实现出色的统计重建，并且在即时预测中的准确性损失最小。