@Article{CiCP-25-947,
author = {Chao Ma, Jianchun Wang and Weinan E},
title = {Model Reduction with Memory and the Machine Learning of Dynamical Systems},
journal = {Communications in Computational Physics},
year = {2018},
volume = {25},
number = {4},
pages = {947--962},
abstract = {<p style="text-align: justify;">The well-known Mori-Zwanzig theory tells us that model reduction leads to
memory effect. For a long time, modeling the memory effect accurately and efficiently
has been an important but nearly impossible task in developing a good reduced model.
In this work, we explore a natural analogy between recurrent neural networks and the
Mori-Zwanzig formalism to establish a systematic approach for developing reduced
models with memory. Two training models —&nbsp;a direct training model and a dynamically coupled training model —&nbsp;are proposed and compared. We apply these methods
to the Kuramoto-Sivashinsky equation and the Navier-Stokes equation. Numerical
experiments show that the proposed method can produce reduced model with good
performance on both short-term prediction and long-term statistical properties.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2018-0269},
url = {http://global-sci.org/intro/article_detail/cicp/12885.html}
}