Volume 25, Issue 4
Model Reduction with Memory and the Machine Learning of Dynamical Systems

Chao Ma, Jianchun Wang & Weinan E

Commun. Comput. Phys., 25 (2019), pp. 947-962.

Published online: 2018-12

[An open-access article; the PDF is free to any online user.]

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  • Abstract

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 — a direct training model and a dynamically coupled training model — 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.

  • Keywords

Model reduction, Mori-Zwanzig, recurrent neural networks.

  • AMS Subject Headings

35Q35, 64P99

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-25-947, author = {}, 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 = {

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 — a direct training model and a dynamically coupled training model — 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0269}, url = {http://global-sci.org/intro/article_detail/cicp/12885.html} }
TY - JOUR T1 - Model Reduction with Memory and the Machine Learning of Dynamical Systems 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 -

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 — a direct training model and a dynamically coupled training model — 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.

Chao Ma, Jianchun Wang & Weinan E. (2020). Model Reduction with Memory and the Machine Learning of Dynamical Systems. Communications in Computational Physics. 25 (4). 947-962. doi:10.4208/cicp.OA-2018-0269
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