Model Reduction with Memory and the Machine Learning of Dynamical Systems (Communications in Computational Physics, 2018)
The well-known Mori-Zwanzig theory tells us that model reduction leads tomemory effect. For a long time, modeling the memory effect accurately and efficientlyhas 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 theMori-Zwanzig formalism to establish a systematic approach for developing reducedmodels with memory. Two training models — a direct training model and a dynamically coupled training model — are proposed and compared. We apply these methodsto the Kuramoto-Sivashinsky equation and the Navier-Stokes equation. Numericalexperiments show that the proposed method can produce reduced model with goodperformance on both short-term prediction and long-term statistical properties.