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<p>Dynamic mode decomposition (DMD), as an equation-free data-driven approach, is widely employed in the construction of reduced-order models (ROMs). However, as an SVD-based approach, DMD faces significant challenges when applied to advection-dominated problems due to the slow decay of its Kolmogorov n-width. This is also a common limitation among many traditional data-driven ROMs. To address these challenges, we introduce an enhanced DMD method based on coordinate transformations, called CT-DMD. In the CT-DMD approach, a mapping is first constructed based on the characteristic lines of the problems and then used to perform coordinate transformation. In the new coordinate system, the translation characteristics in the snapshots can be eliminated. In order to construct a DMD model using the solutions under the transformed coordinate system, interpolation methods are applied to ensure that the transformed solutions align at consistent spatial coordinates. These interpolated solutions can be fed into the DMD algorithm to construct the approximate solutions which should be mapped back to the original coordinate system. Finally, the approximate solutions in the original coordinate can be obtained by the interpolation methods. The good performance of CT-DMD is verified by comparing with that of the standard DMD method through several numerical tests. Additionally, the impact of the number of DMD modes and the interpolation methods on the accuracy of the CT-DMD model is explored in the numerical results.</p>

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An Enhanced Reduced-Order Model Based on Dynamic Mode Decomposition for Advection-Dominated Problems

  • 期刊论文

  • 发表日期：

    2025年2月5日

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  • Yifan Lin,
  • Zhen Gao

摘要

Dynamic mode decomposition (DMD), as an equation-free data-driven approach, is widely employed in the construction of reduced-order models (ROMs). However, as an SVD-based approach, DMD faces significant challenges when applied to advection-dominated problems due to the slow decay of its Kolmogorov n-width. This is also a common limitation among many traditional data-driven ROMs. To address these challenges, we introduce an enhanced DMD method based on coordinate transformations, called CT-DMD. In the CT-DMD approach, a mapping is first constructed based on the characteristic lines of the problems and then used to perform coordinate transformation. In the new coordinate system, the translation characteristics in the snapshots can be eliminated. In order to construct a DMD model using the solutions under the transformed coordinate system, interpolation methods are applied to ensure that the transformed solutions align at consistent spatial coordinates. These interpolated solutions can be fed into the DMD algorithm to construct the approximate solutions which should be mapped back to the original coordinate system. Finally, the approximate solutions in the original coordinate can be obtained by the interpolation methods. The good performance of CT-DMD is verified by comparing with that of the standard DMD method through several numerical tests. Additionally, the impact of the number of DMD modes and the interpolation methods on the accuracy of the CT-DMD model is explored in the numerical results.