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Dynamic mode decomposition with memory.

Ryoji Anzaki1, Kei Sano2, Takuro Tsutsui3

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Summary
This summary is machine-generated.

This study introduces Dynamic Mode Decomposition with Memory (DMDm), a new numerical method to analyze time-series data with memory effects. This approach overcomes limitations of traditional Dynamic Mode Decomposition (DMD), enabling analysis of complex systems.

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Area of Science:

  • Applied Mathematics
  • Data Science
  • Engineering Physics

Background:

  • Dynamic Mode Decomposition (DMD) is a powerful data-driven method for analyzing time-series data.
  • Standard DMD relies on ordinary differential equations, limiting its ability to model systems with memory effects.
  • Memory effects, arising from system-environment interactions, are common in physics and engineering.

Purpose of the Study:

  • To develop a numerical method, Dynamic Mode Decomposition with Memory (DMDm), to analyze multidimensional time-series data incorporating memory effects.
  • To overcome the memoryless restriction inherent in traditional DMD methods.
  • To demonstrate the utility of DMDm in analyzing systems with power-law memory effects using fractional calculus.

Main Methods:

  • Formulation of the abstract algorithmic structure of DMDm.
  • Implementation of DMDm using Caputo fractional derivatives to model power-law memory effects.
  • Development of a fractional DMD method allowing arbitrary-order differential operations.

Main Results:

  • Successful application of the proposed fractional DMD method to synthetic data from fractional oscillators.
  • Accurate estimation of model parameters for systems exhibiting memory effects.
  • Demonstration of DMDm's capability to handle time-series data with memory.

Conclusions:

  • DMDm provides a robust framework for analyzing time-series data with memory effects, extending the capabilities of DMD.
  • The fractional DMD approach enables the modeling of power-law memory phenomena.
  • The proposed method holds significant potential for applications in model estimation, control, and failure detection in mechanical, thermal, and fluid systems, particularly in advanced manufacturing.