# American Institute of Mathematical Sciences

June  2020, 7(1): 35-56. doi: 10.3934/jcd.2020002

## Evaluating the accuracy of the dynamic mode decomposition

 1 Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA 2 Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91125, USA 3 Mechanical Engineering, Florida State University, Tallahassee, FL 32310, USA

* Corresponding author: Clarence W. Rowley

Received  October 2017 Revised  December 2018 Published  November 2019

Fund Project: This work is supported by AFOSR award FA9550-14-1-0289, and DARPA award HR0011-16-C-0116

Dynamic mode decomposition (DMD) gives a practical means of extracting dynamic information from data, in the form of spatial modes and their associated frequencies and growth/decay rates. DMD can be considered as a numerical approximation to the Koopman operator, an infinite-dimensional linear operator defined for (nonlinear) dynamical systems. This work proposes a new criterion to estimate the accuracy of DMD on a mode-by-mode basis, by estimating how closely each individual DMD eigenfunction approximates the corresponding Koopman eigenfunction. This approach does not require any prior knowledge of the system dynamics or the true Koopman spectral decomposition. The method may be applied to extensions of DMD (i.e., extended/kernel DMD), which are applicable to a wider range of problems. The accuracy criterion is first validated against the true error with a synthetic system for which the true Koopman spectral decomposition is known. We next demonstrate how this proposed accuracy criterion can be used to assess the performance of various choices of kernel when using the kernel method for extended DMD. Finally, we show that our proposed method successfully identifies modes of high accuracy when applying DMD to data from experiments in fluids, in particular particle image velocimetry of a cylinder wake and a canonical separated boundary layer.

Citation: Hao Zhang, Scott T. M. Dawson, Clarence W. Rowley, Eric A. Deem, Louis N. Cattafesta. Evaluating the accuracy of the dynamic mode decomposition. Journal of Computational Dynamics, 2020, 7 (1) : 35-56. doi: 10.3934/jcd.2020002
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A diagram summarizing the implementation of the accuracy criterion. Training data is used to approximate Koopman eigenpairs with variants of DMD, while testing data is used to evaluate the quality of Koopman eigenpairs
(A) Analytical eigenvalues. (B) Comparison between the accuracy criterion $\alpha$, eigenvalue error $\tau$, and eigenfunction error $\theta$. The eigenvalues are indexed by their absolute value, in descending order
Eigenfunctions for the system defined in (19), restricted to a domain of $[-1, 1] \times [-1, 1]$, and normalized such that $|\varphi(\mathit{\boldsymbol{x}})|_{max} = 1$. The analytical eigenfunction ${\varphi}_{1, 1}$ shown in (A) is closely approximated by the eigenfunction $\hat \varphi_6$ computed by EDMD, shown in (B). However, the analytical eigenfunction ${\varphi}_{6, 0}$ (with eigenvalue $\mu_{6, 0} = 0.531441$) shown in (C) is not closely approximated by its corresponding eigenfunction $\hat \varphi_{13}$ computed by EDMD (with eigenvalue $\hat\mu_{13} = 0.5250+0.0030j$), whose real part is shown in (D)
Performance of various kernels. Eigenvalue error $\tau$, eigenfunction error $\theta$, and accuracy criterion $\alpha$ are shown. (A) Polynomial kernel of degree $d = 5$, $q = \binom{2+5}{5} = 21$. (B) Exponential kernel, $q = \infty$. (C) Gaussian kernel with $\sigma = 1$, $q = \infty$. (D) Laplacian kernel with $\sigma = 1$, $q = \infty$
Performance of various kernels in the presence of noise. Eigenvalue error $\tau$, eigenfunction error $\theta$, and accuracy criterion $\alpha$ are shown. (A) Polynomial kernel of degree $d = 5$, $q = \binom{2+5}{5} = 21$. (B) Exponential kernel, $q = \infty$. (C) Gaussian kernel with $\sigma = 1$, $q = \infty$. (D) Laplacian kernel with $\sigma = 1$, $q = \infty$
(A) An instantaneous spanwise vorticity field of flow past a circular cylinder at $Re = 413$. (B) Time averaged spanwise vorticity field
(A)-(B) Continuous-time DMD eigenvalues (circles) colored by the accuracy criterion $\alpha$ (A) and mode amplitude $\beta$ (B). Mode amplitudes are normalized by the maximum amplitude. Dominant frequencies (black cross sign $\times$) are shown for comparison. (C)-(E) Three dominant DMD modes (only show real part) picked out by accuracy criterion and mode amplitude
(A)-(B) Continuous-time KDMD eigenvalues (circles) colored by the accuracy criterion $\alpha$ (A) and mode amplitude $\beta$ (B). Mode amplitudes are normalized by the maximum amplitude. Dominant frequencies (black cross sign $\times$) are shown for comparison. (C)-(E) Three dominant KDMD modes (only show real part) picked out by accuracy criterion and mode amplitude
(A) Sketch of the canonical separated flow experiment setup (adapted from [13]). (B) PIV measurement region. (C) Mean spanwise vorticity field
TDMD frequency ($f_{\text{TDMD}}$) and corresponding mode error/amplitude. Mode amplitudes are normalized by the maximum mode amplitude. The truncation level is $r = 25$. The shear layer frequency $f_{\text{SL}} = 106$ Hz is denoted with a red square, and corresponds to the most accurate (smallest $\alpha$) and largest amplitude (largest $\beta$) mode
KDMD frequency ($f_{\text{KDMD}}$) and corresponding mode error/amplitude. The truncation level is $r = 25$. The shear layer frequency $f_{\text{SL}} = 106$ Hz is denoted with a red square
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