The Kernel Perspective on Dynamic Mode Decomposition
Authors: Efrain Gonzalez, Moad Abudia, Michael Jury, Rushikesh Kamalapurkar, Joel A Rosenfeld
TMLR 2024 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | Section 7: Provides a numerical example that compares the developed DMD method to that presented in (Williams et al., 2015b)., Appendix: Includes examples and proofs referenced in the text as well as a few numerical experiments which show that the developed DMD algorithm has nearly identical results to that developed by (Williams et al., 2015b)., Figures 1a and 2 show the ability of the developed DMD technique to reconstruct the training data. The relative reconstruction errors associated with this system are of the order of 1e 8. |
| Researcher Affiliation | Academia | Efrain H Gonzalez EMAIL Department of Mathematics and Statistics University of South Florida Moad Abudia EMAIL School of Mechanical and Aerospace Engineering Oklahoma State University Michael Jury EMAIL Department of Mathematics University of Florida Rushikesh Kamalapurkar EMAIL Department of Mechanical and Aerospace Engineering University of Florida Joel A. Rosenfeld EMAIL Department of Mathematics and Statistics University of South Florida |
| Pseudocode | Yes | Algorithm 1: Pseudocode for the kernel perspective based DMD algorithm. Upon obtaining the Koopman modes, the approximate eigenfunctions, and the eigenvalues, equation 7 is used to compute xi+1. |
| Open Source Code | Yes | The code used to generate these results is publicly available, see Gonzalez et al. (2023)., Efrain Gonzalez, Moad Abudia, Michael Jury, Rushikesh Kamalapurkar, and Joel A. Rosenfeld. https://github.com/scc-lab/publications-code/tree/master/2024-TMLR-Kernel Perspective, 2023. Accessed: 2023-12-29. |
| Open Datasets | Yes | The website (Kutz et al., 2016a) accompanying the textbook (Kutz et al., 2016b) provides several data sets that serve as benchmarks for spectral decomposition approaches to nonlinear modeling, which have been released to the public through their website., Another Numerical example uses a data set that accompanies (Jakob et al., 2020), which consist of flow simulations that range from laminar flow configurations to turbulent flow configurations. The data can be accessed from their website (Jakob et al., 2021). |
| Dataset Splits | Yes | In this demonstration, snapshots 1 through 30 are used as the input data, and snapshots 2 through 31 are used as output data, assuming that the ith and (i + 1)th snapshots satisfy xi+1 = F(xi) for some unknown nonlinear function F. The snapshots are normalized so that the largest 2-norm among all snapshots is 1. To further demonstrate the accuracy of the obtained finite-dimensional representation of the Koopman operator, snapshots 32 through 151 are predicted from snapshot 1 using equation 7. |
| Hardware Specification | No | No specific hardware details (like CPU/GPU models, memory, or cloud instances) are mentioned in the paper for running the experiments. |
| Software Dependencies | No | The paper mentions techniques and integrators used (e.g., 'regularized regression', 'exponential dot product kernel', 'Gaussian RBF kernel', 'fourth order Runge-Kutta integrator'), but does not provide specific software names with version numbers for any libraries, programming languages, or platforms used to implement these methods. |
| Experiment Setup | Yes | DMD is implemented using the exponential dot product kernel, K(x, y) = exp( 1 µx T y) (with µ = 1), and the Gaussian RBF kernel, K(x, y) = exp 1 µ x y 2 2 (with µ = 1)., implemented using regularized regression with ϵ = 10 6 and the exponential dot product kernel with µ = 20) |