Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order models
This work addresses a specific bottleneck in reduced order modeling for parametrized PDEs, offering incremental improvements for computational simulations in fields like engineering and physics.
The paper tackles the problem of poor time extrapolation in deep learning-based reduced order models for nonlinear parametrized dynamical systems by introducing a novel framework that incorporates LSTM cells, enabling long-term prediction up to 15 times beyond the training window and improving testing time performance.
Deep learning-based reduced order models (DL-ROMs) have been recently proposed to overcome common limitations shared by conventional ROMs - built, e.g., exclusively through proper orthogonal decomposition (POD) - when applied to nonlinear time-dependent parametrized PDEs. In particular, POD-DL-ROMs can achieve extreme 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 poor performances regarding time extrapolation tasks. This work aims at taking a further step towards the use of DL algorithms for the efficient numerical approximation of parametrized PDEs by introducing the $μt$-POD-LSTM-ROM framework. This novel technique extends the POD-DL-ROM framework 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 this recurrent architecture enables the extrapolation for time windows up to 15 times larger than the training time domain, and achieves better testing time performances with respect to the already lightning-fast POD-DL-ROMs.