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

Chao Ma ,  Jianchun Wang and Weinan E

10.4208/cicp.OA-2018-0269

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

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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.

  • History

Published online: 2018-12

  • AMS Subject Headings

35Q35, 64P99

  • Cited by