Dynamic Mode Decomposition with Memory

Abstract

In many realms of science and engineering, the time-evolution of a system plays a key role in analyzing, controlling and predicting the behavior of the system. Ranging from the climate data collected by the observation satellites to the classical mechanical motion of manufacturing equipment, there are vast amount of accumulated time-series data in ready-to-use formats.

One established and still developing method for time-series data analysis is the dynamic mode decomposition (DMD), a linear operation-based, model-free method proposed in 2008 by Schmid et al. The DMD, especially in the context of the Koopman theory, assumes the time-evolution model in which the variables evolve in time according to a homogeneous, constant-coefficient first-order ordinary differential equation (ODE). Although the DMD proves itself as one of the successful methods for time-series data analysis in its relatively short history, the abovementioned limitations on the time-evolution model still restricts the DMD from application to wider range of phenomena. Among such phenomena, we pay attention to the memory effects in time-series data. A system is said to have memory if its time-evolution is determined by the past states of the system. The DMD is based on the ODEs and thus cannot incorporate the memory effects.

In this paper, we present a novel method, DMD with memory (DMDm), to overcome the memoryless restriction on the time-evolution model in the existing DMD methods. We introduce a class of initial value problems whose solutions correspond to the eigenmode of a linear operation. Using such a linear transformation in time domain, instead of the time-difference operator, we enlarge the DMD into a wider class of time-evolution models.

As a solid example of this idea, we utilize the Caputo fractional differential to extend the DMD so that one can analyze the time-series data with power-low memory effects, which is seen in various phenomena e.g., viscoelastic matter, fluid dynamics with surface effects, and the mechanical slider with grease. We thus developed fractional DMD, a DMD-based method with arbitrary (real-value) order differential operations. We want to emphasize the fact that you can actually optimize the order of the equation with a standard optimization algorithm, so that one get the order of the equation that best explains the nature of the time-series data. This is also true for any DMDm method with a parameterized time-evolution model. We demonstrated the results for the synthetic data, and successfully estimated the model parameters.

The proposed method is expected to be useful not only for the scientific use, but also for model estimation, control and failure detection of mechanical, thermal and fluid systems in factory machines, such as in modern semiconductor manufacturing equipment.