@article{Fresca2023, 
author = {Stefania Fresca and Federico Fatone and Andrea Manzoni},
title = {Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order models},
year = {2023},
journal = {Mathematics in Engineering},
volume = {5},
number = {6},
pages = {1-36},
keywords = {reduced order modeling, deep learning, proper orthogonal decomposition, long-short term memory networks, time forecasting, parametrized PDEs},
url = {https://www.sciopen.com/article/10.3934/mine.2023096},
doi = {10.3934/mine.2023096},
abstract = {Deep learning-based reduced order models (DL-ROMs) have been recently proposed to overcome common limitations shared by conventional ROMs–built, e.g., through proper orthogonal decomposition (POD)–when applied to nonlinear time-dependent parametrized PDEs. In particular, POD-DL-ROMs can achieve an extremely good efficiency in the training stage and faster than real-time performances at testing, thanks to a prior dimensionality reduction through POD and a DL-based prediction framework. Nonetheless, they share with conventional ROMs unsatisfactory performances regarding time extrapolation tasks. This work aims at taking a further step towards the use of DL algorithms for the efficient approximation of parametrized PDEs by introducing the    μ  t-POD-LSTM-ROM framework. This latter extends the POD-DL-ROMs by adding a two-fold architecture taking advantage of long short-term memory (LSTM) cells, ultimately allowing long-term prediction of complex systems' evolution, with respect to the training window, for unseen input parameter values. Numerical results show that    μ  t-POD-LSTM-ROMs enable the extrapolation for time windows up to 15 times larger than the training time interval, also achieving better performances at testing than POD-DL-ROMs.}
}