TY - JOUR
T1 - Model Reduction with Memory and the Machine Learning of Dynamical Systems
AU - Chao Ma, Jianchun Wang & Weinan E
JO - Communications in Computational Physics
VL - 4
SP - 947
EP - 962
PY - 2018
DA - 2018/12
SN - 25
DO - http://doi.org/10.4208/cicp.OA-2018-0269
UR - https://global-sci.org/intro/article_detail/cicp/12885.html
KW - Model reduction, Mori-Zwanzig, recurrent neural networks.
AB - <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>