The code in this repository features a Python implementation of recurrent neural networks and Koopman-based frameworks for prediction of temporal dynamics of a low-order model of near-wall turbulence proposed by Moehlis et al. (2004, New J. Phys.). The time series generated are used to train long-short-term memory (LSTM) networks and Koopman-based frameworks that can predict the time evolution of the coefficients of the nine-equation model. More details about the implementation and the results are available in "Recurrent neural networks and Koopman-based frameworks for temporal predictions in a low-order model of turbulence", H. Eivazi, L. Guastoni, P. Schlatter, H. Azizpour, R. Vinuesa (2021, International Journal of Heat and Fluid Flow).

This folder contains the data files needed for training and testing of the models.

HDMDs.py: Python class for Hankel dynamic mode decomposition (HDMD).

train_predict_HDMD.py: train an HDMD model and uses it to predict time series based on a given input seed and reproduce the long-term statistics.

KNFs.py: Python class for Koopman-based framework with nonlinear forcing (KNF).

train_predict_KNF.py: train a KNF model and uses it to predict time series based on a given input seed and reproduce the long-term statistics.

LSTM1_t10000.h5: a trained LSTM model. More details in "Predictions of turbulent shear flows using deep neural networks", P.A. Srinivasan, L. Guastoni, H. Azizpour, P. Schlatter, R. Vinuesa (2019, Phys. Rev. Fluids; also available in arXiv)

predict_LSTM.py: uses the trained LSTM model to predict time series based on a given input seed and reproduce the long-term statistics.

lstm_pred_func.py: Python function for the LSTM prediction iterations.

compare_short_term_predictions.py: compares the performance of KNF, HDMD, and LSTM models in the short-term predictions by producing Figures 2 and 3 of the paper.

compare_long_term_statistics.py: compares the performance of KNF, HDMD, and LSTM models in the reproduction of the long-term statistics by producing Figure 5 of the paper.

Moehlis_perturb_generator.py: allows to generate time series that have a perturbation at time t=500.

Lyapunov_exponents.py: computes the Lyapunov exponents of these time series and compares it with the LSTM predictions.

Results of the prediction of the time series and reproduction of the long-term statistics will be stored in this directory.

optht.py: Optimal hard threshold for singular values.

statistics.py: a Python function that computes the long-term statistics from given reference and predicted time series.