Model Reduction with Memory and the Machine Learning of Dynamical Systems
This work addresses the long-standing problem of efficiently capturing memory effects in reduced models for dynamical systems, which is crucial for applications in physics and engineering, though it appears incremental by building on existing theory with machine learning techniques.
The authors tackled the challenge of accurately modeling memory effects in reduced dynamical systems by drawing an analogy between recurrent neural networks and the Mori-Zwanzig formalism, proposing two training models that demonstrated good performance in short-term prediction and long-term statistical properties for the Kuramoto-Sivashinsky and Navier-Stokes equations.
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.