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Dynamic Mode Decomposition with Memory

##article.authors##

  • Ryoji Anzaki Advanced Engineering 1st Department, Digital Design Center, Tokyo Electron Ltd. https://orcid.org/0000-0002-6395-8799
  • Kei Sano Advanced Engineering 1st Department, Digital Design Center, Tokyo Electron Ltd.
  • Takuro Tsutsui Advanced Engineering 2nd Department, Digital Design Center, Tokyo Electron Ltd. https://orcid.org/0000-0001-9683-3952
  • Masato Kazui Advanced Engineering 1st Department, Digital Design Center, Tokyo Electron Ltd.
  • Takahito Matsuzawa Advanced Engineering 1st Department, Digital Design Center, Tokyo Electron Ltd. https://orcid.org/0000-0001-5430-1501

DOI:

https://doi.org/10.51094/jxiv.176

キーワード:

dynamic mode decomposition、 fractional-order derivative、 time-series data

抄録

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.

利益相反に関する開示

We declare that we have no conflicts of interest.

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著者の経歴

Ryoji Anzaki、Advanced Engineering 1st Department, Digital Design Center, Tokyo Electron Ltd.

2016-2020: Graduate School of Engineering, The University of Tokyo (PhD)

2020-2021: Project Researcher, Earthquake Research Institute, The University of Tokyo

2021-Current: Scientist, Tokyo Electron Ltd.

引用文献

T. Askham and J. N. Kutz, Variable projection methods for an optimized dynamic mode decomposition, SIAM Journal on Applied Dynamical Systems, 17 (2018), pp. 380–416.

S. L. Brunton, M. Budišić, E. Kaiser, and J. N. Kutz, Modern Koopman theory for dynamical systems, arXiv preprint arXiv:2102.12086, (2021).

S. Burov and E. Barkai, Fractional Langevin equation: Overdamped, underdamped, and critical behaviors, Physical Review E, 78 (2008), p. 031112.

H. J. Haubold, A. M. Mathai, and R. K. Saxena, Mittag-Leffler functions and their applications, Journal of applied mathematics, 2011 (2011).

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, vol. 204, elsevier, (2006).

D. Matignon, Stability results for fractional differential equations with applications to control processing, in Computational engineering in systems applications, vol. 2, Citeseer, 1996, pp. 963–968.

I. Mezić, Spectral properties of dynamical systems, model reduction and decompositions, Nonlinear Dynamics, 41 (2005), pp. 309–325.

I. Mezic, Analysis of fluid flows via spectral properties of the Koopman operator, Annual Review of Fluid Mechanics, 45 (2013), pp. 357–378.

I. Podlubny, Fractional-order systems and pi/sup /spl lambda//d/sup /spl mu//-controllers, IEEE Transactions on Automatic Control, 44 (1999), pp. 208–214.

J. L. Proctor, S. L. Brunton, and J. N. Kutz, Dynamic mode decomposition with control, SIAM Journal on Applied Dynamical Systems, 15 (2016), pp. 142–161.

E. Rodrigues, B. Zadrozny, C. Watson, and D. Gold, Decadal forecasts with resdmd: a residual DMD neural network, arXiv preprint arXiv:2106.11111, (2021).

C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, and D. S. Henningson, Spectral analysis of nonlinear flows, Journal of fluid mechanics, 641 (2009), pp. 115–127.

I. Sakata, T. Sakata, K. Mizoguchi, S. Tanaka, G. Oohata, I. Akai, Y. Igarashi, Y. Nagano, and M. Okada, Complex energies of the coherent longitudinal optical phonon–plasmon coupled mode according to dynamic mode decomposition analysis, Scientific Reports, 11 (2021), pp. 1–10.

A. A. K. Samko, Stefan G. and O. I. Marichev, Fractional integrals and derivatives: theory and applications, Gordon and Breach Science Publishers, 1993.

M. Sasaki, Y. Kawachi, R. Dendy, H. Arakawa, N. Kasuya, F. Kin, K. Yamasaki, and S. Inagaki, Using dynamical mode decomposition to extract the limit cycle dynamics of modulated turbulence in a plasma simulation, Plasma Physics and Controlled Fusion, 61 (2019), p. 112001.

D. Sashidhar and J. N. Kutz, Bagging, optimized dynamic mode decomposition for robust, stable forecasting with spatial and temporal uncertainty quantification, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 380 (2022), p. 20210199.

P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, Journal of fluid mechanics, 656 (2010), pp. 5–28.

P. J. Schmid, Dynamic mode decomposition and its variants, Annual Review of Fluid Mechanics, 54 (2022), pp. 225–254.

A. A. Stanislavsky, Fractional oscillator, Phys. Rev. E, 70 (2004), p. 051103.

N. Sugimoto and T. Horioka, Dispersion characteristics of sound waves in a tunnel with an array of Helmholtz resonators, Journal of the Acoustical Society of America, 97 (1995), pp. 1446–1459.

A. Svenkeson, B. Glaz, S. Stanton, and B. J. West, Spectral decomposition of nonlinear systems with memory, Phys. Rev. E, 93 (2016), p. 022211.

V. E. Tarasov, General fractional dynamics, Mathematics, 9 (2021), p. 1464.

J. Tenreiro Machado and A. Azenha, Fractional-order hybrid control of robot manipulators, in SMC’98 Conference Proceedings. 1998 IEEE International Conference on Systems, Man, and Cybernetics (Cat. No.98CH36218), vol. 1, 1998, pp. 788–793 vol.1.

J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton, and J. N. Kutz, On dynamic mode decomposition: Theory and applications, Journal of Computational Dynamics, 1 (2014), pp. 391–421.

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel. prog fract differ appl 1 (2): 73–85 (2015).

A. Baba and S. Adachi, Simultaneous state of charge and parameter estimation of lithium-ion battery using log-normalized unscented kalman filter, in 2015 American Control Conference (ACC) (2015) pp. 311–316.

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投稿日時: 2022-11-16 02:50:15 UTC

公開日時: 2022-11-17 02:11:15 UTC — 2023-08-24 00:33:48 UTCに更新

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改版理由

1. We added numerical results where the number of observations is smaller than the degree of freedom. 2. We modified the affiliations because our department has been reorganized.
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