StOKeDMD: Streaming Occupation kernel dynamic mode decomposition

Efrain H. Gonzalez; Benjamin P. Russo; M. Paul Laiu; Richard Archibald

doi:10.3934/ammc.2024022

Article Contents

2024, Volume 2, Issue 4: 433-464. Doi: 10.3934/ammc.2024022

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StOKeDMD: Streaming Occupation kernel dynamic mode decomposition

1.
Sandia National Laboratories, Albuquerque, NM 87123, USA
2.
Riverside Research, New York, NY 10038, USA
3.
Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

^*Corresponding author: Efrain H. Gonzalez
^*Corresponding author: Efrain H. Gonzalez

Received: July 2024

Revised: November 2024

Early access: December 2024

Published: December 2024

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Abstract

Dynamic mode decomposition (DMD) has become a common technique for constructing surrogate models for dynamical systems from observed system states. The Occupation Kernel DMD (OKDMD) method proposed in (Rosenfeld et al., 2022) and (Rosenfeld et al., 2024) is a Liouville operator based method that builds surrogate models from system state trajectories. This paper proposes an extension of OKDMD to the case when the system states are observed in a streaming fashion, i.e., only a small fraction of the state trajectory is available at a given time. The developed method, Streaming Occupation Kernel DMD (StOKeDMD), accommodates the streaming data input by leveraging properties of specific choices of kernel functions and occupation kernels. We apply the StoKeDMD method as a compression method for streaming data, analyze the memory complexity, and demonstrate the performance of StoKeDMD in the compression of streaming data generated from a Lorenz system and a fluid flow simulation.

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Mathematics Subject Classification: 46E22, 37Mxx.

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Figure 1. Figure 1A plots 12 trajectories from the Lorenz system (51) over the interval $ [0, 0.1]$. Figures 1B and 1C show a reconstruction of these trajectories using the StOKeDMD algorithm

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Figure 2. Figure 2A shows the percent error between Figure 1A and Figure 1B. Figure 2B shows the percent error between 1A and Figure 1C. For reference, the percent error for reconstructing the $ 12 $ trajectories using offline OKDMD is displayed in Figure 2C

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Figure 3. Figure 3A shows an example of the $ 12 $ Lorenz trajectories perturbed by noise. Figures 3B and 3C show the reconstuctions of the trajectories using a StOKeDMD model constructed from the data in Figure 3A

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Figure 4. Figures 4A and 4B show the average percent error between the $ 1000 $ systems and the $ 12 $ trajectories in 1A using a StOKeDMD model constructed from the data

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Figure 5. Figure 5A plots the $ 12 $ reconstructed trajectories from the Lorenz system (51) over the interval $ [0, 0.1] $ based of the model obtained from the original DMD algorithm. Figure 5B shows the percent error between 1A and 5A

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Figure 6. Figure 6A plots the $ 12 $ new trajectories from the Lorenz system (51) over the interval $ [0, 0.1] $. Figure 6B shows a prediction of these trajectories using the StOKeDMD model developed from the original $ 12 $ trajectories presented in Figure 1A and initial values which correspond to those of the $ 12 $ new trajectories. Figure 6C shows the prediction percent error between 6A and 6B

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Figure 7. Figure 7A plots the reconstruction of the $ 12 $ trajectories from the Lorenz system (51) over the interval $ [0, 0.1] $ using the Hyper StOKeD algorithm. Figure 7B shows the percent error for reconstructing the $ 12 $ trajectories when using the Hyper StOKeD algorithm

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Figure 8. Figures 8A, 8D, 8G, and 8J show the true snapshots at time $ 0.18 $, $ 1.58 $, $ 2.24 $, and $ 2.98 $, respectively. Figures 8B, 8E, 8H, and 8K show the reconstruction of the snapshots at time $ 0.18 $, $ 1.58 $, $ 2.24 $, and $ 2.98 $, using the DMD method. Figures 8C, 8F, 8I, and 8L show the reconstruction of the snapshots at time $ 0.18 $, $ 1.58 $, $ 2.24 $, and $ 2.98 $ using the StOKeDMD method

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Figure 9. Figures 9A and 9B show the percent error between the true trajectory represented by $ {\mathbf A} $ and the trajectories generated by using a StOKeDMD model constructed from data set $ {\mathbf B} $ and data set $ {\mathbf C} $, respectively

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Figure 10. Figures 10A and 10B show the average percent error between the true trajectory represented by $ {\mathbf A} $ and the trajectories generated by using a StOKeDMD model constructed from data set $ {\mathbf B} $ and data set $ {\mathbf C} $, respectively

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Figure 11. Figure 11A displays the values obtained from the Gaussian RBF kernel as well as its approximations over the points on the selected grid. Figure 11B shows the percent error obtained as a result of approximating the Gaussian RBF kernel with a $ 3^{rd} $ and $ 4^{th} $ degree Hermite expansion

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Table 1. This table displays the state dimension, number of trajectories, and length of each trajectory found in the three data sets that were used for Experiment $ 7 $

Experiment $ 7 $ Data Sets
Data Set	State Dimension	Trajectory Length	# of Trajectories
$ {\mathbf A} $	$ 16000 $	$ 101 $	$ 100 $
$ {\mathbf B} $	$ 16000 $	$ 11 $	$ 100 $
$ {\mathbf C} $	$ 16000 $	$ 6 $	$ 100 $

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Table 2. In the above table, memory requirements refers to the number of double precision elements that must be stored in memory. Additionally the table lists the memory requirements associated with the algorithms used in each experiment as well as the compression ratio associated with using any of the listed algorithms. In the case of Experiment $ 7 $ the inequality is indicative of the fact that several data sets were tested whereas in the case of Experiments $ 1 $, $ 2 $, and $ 4 $ the inequality is indicative of the fact that the algorithm was tested with two Hermite expansions of different degrees. The final row of the table shows the offline computational complexity for each algorithm. An "or" statement is used to address the potential scenario where $ \hat L $ or $ W $ is larger than the number of trajectories, $ M $. In the case of OKDMD, it is possible for $ S \times N^2 $ to be larger than $ M^3 $

Comparison of Working Memory Requirements For Each Experiment
Experiments	OKDMD Requirement	StOKeDMD Requirement	Hyper StOKeD Requirement	Compression Ratio
$ 1 $, $ 2 $, & $ 4 $	$ 3636 $	$ \leq 876 $		$ \sim 18.94\text{x} $
$ 6 $	$ 151 \times 89351 $	$ 588 \times 89351 $		$ \sim 1.03\text{x} $
$ 7 $	$ \geq 501 \times 16000 $	$ 400 \times 16000 $		$ \geq 4.97\text{x} $
$ 5 $	$ 3636 $		$ 70488 $	$ \sim 18.94\text{x} $
Computational Complexity	$ O(M^3) $ or $ O(S\times N^2) $	$ O(M^3) $ or $ O(\hat L \times M^2) $	$ O(M^3) $ or $ O(W\times M^2) $

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