Dynamic Mode Decomposition with Memory

Abstract

This study proposed a novel method of dynamic mode decomposition with memory (DMDm) to analyze multi-dimensional time-series data with memory effects. The memory effect is a widely observed phenomenon in physics and engineering and is considered to be the result of interactions between the system and environment. Dynamic mode decomposition (DMD) is a linear operation-based, model-free method for multi-dimensional time-series data proposed in 2008. Although DMD is a successful method for time-series data analysis, it is based on ordinary differential equations and thus, cannot incorporate memory effects. In this study, we formulated the abstract algorithmic structure of DMDm and demonstrate its utility in overcoming the memoryless restriction imposed by existing DMD methods on the time-evolution model. In the numerical demonstration, we utilized the Caputo fractional differential to implement an example of DMDm such that the time-series data could be analyzed with power-law memory effects. Thus, we developed a fractional DMD, which is a DMD-based method with arbitrary (real value) order differential operations. The proposed method was applied to synthetic data from a set of fractional oscillators and model parameters were estimated successfully. The proposed method is expected to be useful for scientific applications, and aid in model estimation, control, and failure detection of mechanical, thermal, and fluid systems in factory machines, such as modern semiconductor manufacturing equipment.