Optimal Prediction and the Dynamic-Mode Decomposition

Over the last twenty years, the Dynamic Mode Decomposition (DMD) has become a standard tool in performing data analysis and model generation within the fluid mechanics community. A standout feature of DMD is that there is no need for prior modeling and thus it runs on data alone. However, this makes resolving issues stemming from unresolved or missing data critical to its future use. In this talk then, we present a novel extension to the standard DMD whereby memory effects due to unresolved degrees of freedom in data can be accounted for using the Mori-Zwanzig (MZ) formalism from non-equilibrium statistical mechanics. MZ based methods have been finding a range of uses in engineering contexts where measurements are sparse. Thus our augmented, memory dependent DMD, or MDDMD, method allows for using coarse-grained data to generate more sophisticated and accurate data-driven models. We will present a brief introduction to the Mori-Zwanzig formalism, an outline of how this allows us to include memory effects in the DMD framework, and then an application of our method to a model problem from the now seminal paper of Chorin, Hald, and Kupferman. Some technical dilemmas and directions for future research will also be discussed.