Model Reduction with Memory and the Machine Learning of Dynamical Systems

Model Reduction with Memory and the Machine Learning of Dynamical Systems Chao Ma, Jianchun Wang †, Weinan E ‡ 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. In science and engineering, many high-dimensional dynamical systems are too complicated to solve in detail. Nor is it necessary since usually we are only interested in a small subset of the variables representing the gross behavior of the system. Therefore, it is useful to develop reduced models which can approximate the variables of interest without solving the full system. This is the celebrated model reduction problem.