An Adaptive Method Based on Local Dynamic Mode Decomposition for Parametric Dynamical Systems

An Adaptive Method Based on Local Dynamic Mode Decomposition for Parametric Dynamical Systems

Year:    2024

Author:    Qiuqi Li, Chang Liu, Mengnan Li, Pingwen Zhang

Communications in Computational Physics, Vol. 35 (2024), Iss. 1 : pp. 38–69

Abstract

Parametric dynamical systems are widely used to model physical systems, but their numerical simulation can be computationally demanding due to nonlinearity, long-time simulation, and multi-query requirements. Model reduction methods aim to reduce computation complexity and improve simulation efficiency. However, traditional model reduction methods are inefficient for parametric dynamical systems with nonlinear structures. To address this challenge, we propose an adaptive method based on local dynamic mode decomposition (DMD) to construct an efficient and reliable surrogate model. We propose an improved greedy algorithm to generate the atoms set $\Theta$ based on a sequence of relatively small training sets, which could reduce the effect of large training set. At each enrichment step, we construct a local sub-surrogate model using the Taylor expansion and DMD, resulting in the ability to predict the state at any time without solving the original dynamical system. Moreover, our method provides the best approximation almost everywhere over the parameter domain with certain smoothness assumptions, thanks to the gradient information. At last, three concrete examples are presented to illustrate the effectiveness of the proposed method.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2023-0163

Communications in Computational Physics, Vol. 35 (2024), Iss. 1 : pp. 38–69

Published online:    2024-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    32

Keywords:    Parametric dynamical systems dynamic mode decomposition greedy algorithm Taylor expansion surrogate model.

Author Details

Qiuqi Li

Chang Liu

Mengnan Li

Pingwen Zhang