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
References:
[1]

T. Askham and J. N. Kutz, Variable projection methods for an optimized dynamic mode decomposition, SIAM J. Appl. Dyn. Syst., 17 (2018), 380-416.  doi: 10.1137/M1124176.  Google Scholar

[2]

S. Bagheri, Effects of weak noise on oscillating flows: Linking quality factor, Floquet modes, and Koopman spectrum, Phys. Fluids, 26 (2014). doi: 10.1063/1.4895898.  Google Scholar

[3]

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K. K. ChenJ. H. Tu and C. W. Rowley, Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses, J. Nonlinear Sci., 22 (2012), 887-915.  doi: 10.1007/s00332-012-9130-9.  Google Scholar

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A. Cotter, J. Keshet and N. Srebro, Explicit approximations of the Gaussian kernel, preprint, arXiv: 1109.4603. Google Scholar

[6]

S. T. M. DawsonM. S. HematiM. O. Williams and C. W. Rowley, Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition, Experiments in Fluids, 3 (2016), 1-19.  doi: 10.1007/s00348-016-2127-7.  Google Scholar

[7]

E. Deem, L. Cattafesta, H. Zhang, C. Rowley, M. Hemati, F. Cadieux and R. Mittal, Identifying dynamic modes of separated flow subject to ZNMF-based control from surface pressure measurements, 47th AIAA Fluid Dynamics Conference, 2017. doi: 10.2514/6.2017-3309.  Google Scholar

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Z. Drmač, I. Mezić and R. Mohr, Data driven modal decompositions: analysis and enhancements, SIAM J. Sci. Comput., 40 (2018), A2253–A2285. doi: 10.1137/17M1144155.  Google Scholar

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G. E. Fasshauer, Positive definite kernels: Past, present and future, Dolomite Research Notes on Approximation, 4 (2011), 21-63.   Google Scholar

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[13]

J. Griffin, M. Oyarzun, L. N. Cattafesta, J. H. Tu, C. W. Rowley and R. Mittal, Control of a canonical separated flow, 43rd AIAA Fluid Dynamics Conference, 2013. doi: 10.2514/6.2013-2968.  Google Scholar

[14]

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[16]

M. S. HematiC. W. RowleyE. A. Deem and L. N. Cattafesta, De-biasing the dynamic mode decomposition for applied Koopman spectral analysis of noisy datasets, Theoretical and Computational Fluid Dynamics, 31 (2017), 349-368.  doi: 10.1007/s00162-017-0432-2.  Google Scholar

[17]

M. S. Hemati, M. O. Williams and C. W. Rowley, Dynamic mode decomposition for large and streaming datasets, Physics of Fluids, 26 (2014). doi: 10.1063/1.4901016.  Google Scholar

[18]

M. R. Jovanović, P. J. Schmid and J. W. Nichols, Sparsity-promoting dynamic mode decomposition, Physics of Fluids, 26 (2014). doi: 10.1063/1.4863670.  Google Scholar

[19]

J. Kou and W. Zhang, An improved criterion to select dominant modes from dynamic mode decomposition, Eur. J. Mech. B Fluids, 62 (2017), 109-129.  doi: 10.1016/j.euromechflu.2016.11.015.  Google Scholar

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[22]

B. R. NoackW. StankiewiczM. Morzyński and P. J. Schmid, Recursive dynamic mode decomposition of transient and post-transient wake flows, J. Fluid Mech., 809 (2016), 843-872.  doi: 10.1017/jfm.2016.678.  Google Scholar

[23]

C. W. Rowley and S. T. Dawson, Model reduction for flow analysis and control, Annual Review Fluid Mech., 49 (2017), 387-417.  doi: 10.1146/annurev-fluid-010816-060042.  Google Scholar

[24]

C. W. RowleyI. MezićS. BagheriP. Schlatter and D. S. Henningson, Spectral analysis of nonlinear flows, J. Fluid Mech., 641 (2009), 115-127.  doi: 10.1017/S0022112009992059.  Google Scholar

[25]

P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656 (2010), 5-28.  doi: 10.1017/S0022112010001217.  Google Scholar

[26]

C. ScovelD. HushI. Steinwart and J. Theiler, Radial kernels and their reproducing kernel Hilbert spaces, J. Complexity, 26 (2010), 641-660.  doi: 10.1016/j.jco.2010.03.002.  Google Scholar

[27]

C. R. Souza, Kernel functions for machine learning applications, Creative Commons Attribution-Noncommercial-Share Alike, 3. Google Scholar

[28]

J. H. TuC. W. RowleyJ. N. Kutz and J. K. Shang, Spectral analysis of fluid flows using sub-Nyquist-rate PIV data, Experiments in Fluids, 55 (2014), 1-13.  doi: 10.1007/s00348-014-1805-6.  Google Scholar

[29]

J. H. TuC. W. RowleyD. M. LuchtenburgS. L. Brunton and J. N. Kutz, On dynamic mode decomposition: Theory and applications, J. Comput. Dyn., 1 (2014), 391-421.  doi: 10.3934/jcd.2014.1.391.  Google Scholar

[30]

M. O. WilliamsI. G. Kevrekidis and C. W. Rowley, A data–driven approximation of the Koopman operator: Extending dynamic mode decomposition, J. Nonlinear Sci., 25 (2015), 1307-1346.  doi: 10.1007/s00332-015-9258-5.  Google Scholar

[31]

M. O. WilliamsC. W. Rowley and I. G. Kevrekidis, A kernel-based method for data-driven Koopman spectral analysis, J. Comput. Dyn., 2 (2015), 247-265.  doi: 10.3934/jcd.2015005.  Google Scholar

[32]

A. WynnD. PearsonB. Ganapathisubramani and P. Goulart, Optimal mode decomposition for unsteady flows, J. Fluid Mech., 733 (2013), 473-503.  doi: 10.1017/jfm.2013.426.  Google Scholar

[33]

H. ZhangC. W. RowleyE. A. Deem and L. N. Cattafesta, Online dynamic mode decomposition for time-varying systems, SIAM J. Appl. Dyn. Syst., 18 (2019), 1586-1609.  doi: 10.1137/18M1192329.  Google Scholar

show all references

References:
[1]

T. Askham and J. N. Kutz, Variable projection methods for an optimized dynamic mode decomposition, SIAM J. Appl. Dyn. Syst., 17 (2018), 380-416.  doi: 10.1137/M1124176.  Google Scholar

[2]

S. Bagheri, Effects of weak noise on oscillating flows: Linking quality factor, Floquet modes, and Koopman spectrum, Phys. Fluids, 26 (2014). doi: 10.1063/1.4895898.  Google Scholar

[3]

C. M. Bishop, Pattern Recognition and Machine Learning, Information Science and Statistics, Springer, New York, 2006.  Google Scholar

[4]

K. K. ChenJ. H. Tu and C. W. Rowley, Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses, J. Nonlinear Sci., 22 (2012), 887-915.  doi: 10.1007/s00332-012-9130-9.  Google Scholar

[5]

A. Cotter, J. Keshet and N. Srebro, Explicit approximations of the Gaussian kernel, preprint, arXiv: 1109.4603. Google Scholar

[6]

S. T. M. DawsonM. S. HematiM. O. Williams and C. W. Rowley, Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition, Experiments in Fluids, 3 (2016), 1-19.  doi: 10.1007/s00348-016-2127-7.  Google Scholar

[7]

E. Deem, L. Cattafesta, H. Zhang, C. Rowley, M. Hemati, F. Cadieux and R. Mittal, Identifying dynamic modes of separated flow subject to ZNMF-based control from surface pressure measurements, 47th AIAA Fluid Dynamics Conference, 2017. doi: 10.2514/6.2017-3309.  Google Scholar

[8]

Z. Drmač, I. Mezić and R. Mohr, Data driven modal decompositions: analysis and enhancements, SIAM J. Sci. Comput., 40 (2018), A2253–A2285. doi: 10.1137/17M1144155.  Google Scholar

[9]

D. DukeJ. Soria and D. Honnery, An error analysis of the dynamic mode decomposition, Experiments in Fluids, 52 (2012), 529-542.  doi: 10.1007/s00348-011-1235-7.  Google Scholar

[10]

G. E. Fasshauer, Positive definite kernels: Past, present and future, Dolomite Research Notes on Approximation, 4 (2011), 21-63.   Google Scholar

[11] G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013.   Google Scholar
[12]

A. Gretton, Introduction to RKHS, and some simple kernel algorithms, Advanced Topics in Machine Learning, Lecture at University College London. Google Scholar

[13]

J. Griffin, M. Oyarzun, L. N. Cattafesta, J. H. Tu, C. W. Rowley and R. Mittal, Control of a canonical separated flow, 43rd AIAA Fluid Dynamics Conference, 2013. doi: 10.2514/6.2013-2968.  Google Scholar

[14]

D. M. Hawkins, The problem of overfitting, J. Chem. Information Comput. Sci., 44 (2004), 1-12.  doi: 10.1021/ci0342472.  Google Scholar

[15]

M. S. Hemati, E. A. Deem, M. O. Williams, C. W. Rowley and L. N. Cattafesta, Improving separation control with noise-robust variants of dynamic mode decomposition, 54th AIAA Aerospace Sciences Meeting, 2016. doi: 10.2514/6.2016-1103.  Google Scholar

[16]

M. S. HematiC. W. RowleyE. A. Deem and L. N. Cattafesta, De-biasing the dynamic mode decomposition for applied Koopman spectral analysis of noisy datasets, Theoretical and Computational Fluid Dynamics, 31 (2017), 349-368.  doi: 10.1007/s00162-017-0432-2.  Google Scholar

[17]

M. S. Hemati, M. O. Williams and C. W. Rowley, Dynamic mode decomposition for large and streaming datasets, Physics of Fluids, 26 (2014). doi: 10.1063/1.4901016.  Google Scholar

[18]

M. R. Jovanović, P. J. Schmid and J. W. Nichols, Sparsity-promoting dynamic mode decomposition, Physics of Fluids, 26 (2014). doi: 10.1063/1.4863670.  Google Scholar

[19]

J. Kou and W. Zhang, An improved criterion to select dominant modes from dynamic mode decomposition, Eur. J. Mech. B Fluids, 62 (2017), 109-129.  doi: 10.1016/j.euromechflu.2016.11.015.  Google Scholar

[20]

J. N. Kutz, S. L. Brunton, B. W. Brunton and J. L. Proctor, Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems, 149, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974508.  Google Scholar

[21]

J. Mercer, Functions of positive and negative type, and their connection with the theory of integral equations, Philosophical Transac. Royal Soc. London, Ser. A, 209 (1909), 415-446.  doi: 10.1098/rsta.1909.0016.  Google Scholar

[22]

B. R. NoackW. StankiewiczM. Morzyński and P. J. Schmid, Recursive dynamic mode decomposition of transient and post-transient wake flows, J. Fluid Mech., 809 (2016), 843-872.  doi: 10.1017/jfm.2016.678.  Google Scholar

[23]

C. W. Rowley and S. T. Dawson, Model reduction for flow analysis and control, Annual Review Fluid Mech., 49 (2017), 387-417.  doi: 10.1146/annurev-fluid-010816-060042.  Google Scholar

[24]

C. W. RowleyI. MezićS. BagheriP. Schlatter and D. S. Henningson, Spectral analysis of nonlinear flows, J. Fluid Mech., 641 (2009), 115-127.  doi: 10.1017/S0022112009992059.  Google Scholar

[25]

P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656 (2010), 5-28.  doi: 10.1017/S0022112010001217.  Google Scholar

[26]

C. ScovelD. HushI. Steinwart and J. Theiler, Radial kernels and their reproducing kernel Hilbert spaces, J. Complexity, 26 (2010), 641-660.  doi: 10.1016/j.jco.2010.03.002.  Google Scholar

[27]

C. R. Souza, Kernel functions for machine learning applications, Creative Commons Attribution-Noncommercial-Share Alike, 3. Google Scholar

[28]

J. H. TuC. W. RowleyJ. N. Kutz and J. K. Shang, Spectral analysis of fluid flows using sub-Nyquist-rate PIV data, Experiments in Fluids, 55 (2014), 1-13.  doi: 10.1007/s00348-014-1805-6.  Google Scholar

[29]

J. H. TuC. W. RowleyD. M. LuchtenburgS. L. Brunton and J. N. Kutz, On dynamic mode decomposition: Theory and applications, J. Comput. Dyn., 1 (2014), 391-421.  doi: 10.3934/jcd.2014.1.391.  Google Scholar

[30]

M. O. WilliamsI. G. Kevrekidis and C. W. Rowley, A data–driven approximation of the Koopman operator: Extending dynamic mode decomposition, J. Nonlinear Sci., 25 (2015), 1307-1346.  doi: 10.1007/s00332-015-9258-5.  Google Scholar

[31]

M. O. WilliamsC. W. Rowley and I. G. Kevrekidis, A kernel-based method for data-driven Koopman spectral analysis, J. Comput. Dyn., 2 (2015), 247-265.  doi: 10.3934/jcd.2015005.  Google Scholar

[32]

A. WynnD. PearsonB. Ganapathisubramani and P. Goulart, Optimal mode decomposition for unsteady flows, J. Fluid Mech., 733 (2013), 473-503.  doi: 10.1017/jfm.2013.426.  Google Scholar

[33]

H. ZhangC. W. RowleyE. A. Deem and L. N. Cattafesta, Online dynamic mode decomposition for time-varying systems, SIAM J. Appl. Dyn. Syst., 18 (2019), 1586-1609.  doi: 10.1137/18M1192329.  Google Scholar

Figure 1.  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
Figure 2.  (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
Figure 3.  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)
Figure 4.  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 $
Figure 5.  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 $
Figure 6.  (A) An instantaneous spanwise vorticity field of flow past a circular cylinder at $ Re = 413 $. (B) Time averaged spanwise vorticity field
Figure 7.  (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
Figure 8.  (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 $) are shown for comparison. (C)-(E) Three dominant KDMD modes (only show real part) picked out by accuracy criterion and mode amplitude
Figure 9.  (A) Sketch of the canonical separated flow experiment setup (adapted from [13]). (B) PIV measurement region. (C) Mean spanwise vorticity field
Figure 10.  TDMD frequency ($ 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 $) mode
Figure 11.  KDMD frequency ($ 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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