Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
139 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Data-driven discovery with Limited Data Acquisition for fluid flow across cylinder (2312.12630v1)

Published 19 Dec 2023 in math.DS, cs.LG, math.CV, math.FA, and math.SP

Abstract: One of the central challenge for extracting governing principles of dynamical system via Dynamic Mode Decomposition (DMD) is about the limit data availability or formally called as Limited Data Acquisition in the present paper. In the interest of discovering the governing principles for a dynamical system with limited data acquisition, we provide a variant of Kernelized Extended DMD (KeDMD) based on the Koopman operator which employ the notion of Gaussian random matrix to recover the dominant Koopman modes for the standard fluid flow across cylinder experiment. It turns out that the traditional kernel function, Gaussian Radial Basis Function Kernel, unfortunately, is not able to generate the desired Koopman modes in the scenario of executing KeDMD with limited data acquisition. However, the Laplacian Kernel Function successfully generates the desired Koopman modes when limited data is provided in terms of data-set snapshot for the aforementioned experiment and this manuscripts serves the purpose of reporting these exciting experimental insights. This paper also explores the functionality of the Koopman operator when it interacts with the reproducing kernel Hilbert space (RKHS) that arises from the normalized probability Lebesgue measure $d\mu_{\sigma,1,\mathbb{C}n}(z)=(2\pi\sigma2){-n}\exp\left(-\frac{|z|_2}{\sigma}\right)dV(z)$ when it is embedded in $L2-$sense for the holomorphic functions over $\mathbb{C}n$, in the aim of determining the Koopman modes for fluid flow across cylinder experiment. We explore the operator-theoretic characterizations of the Koopman operator on the RKHS generated by the normalized Laplacian measure $d\mu_{\sigma,1,\mathbb{C}n}(z)$ in the $L2-$sense. In doing so, we provide the compactification & closable characterization of Koopman operator over the RKHS generated by the normalized Laplacian measure in the $L2-$sense.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (108)
  1. Operator-theoretic framework for forecasting nonlinear time series with kernel analog techniques. Physica D: Nonlinear Phenomena 409 (2020), 132520.
  2. Aronszajn, N. Theory of reproducing kernels. Transactions of the American Mathematical Society 68, 3 (1950), 337–404.
  3. Kernel learning for robust Dynamic Mode Decomposition: Linear and Nonlinear disambiguation optimization. Proceedings of the Royal Society A 478, 2260 (2022), 20210830.
  4. Bagheri, S. Koopman-mode decomposition of the cylinder wake. Journal of Fluid Mechanics 726 (2013), 596–623.
  5. Bayart, F. Parabolic composition operators on the ball. Advances in Mathematics 223, 5 (2010), 1666–1705.
  6. Bayart, F. Composition operators on the polydisk induced by affine maps. Journal of Functional Analysis 260, 7 (2011), 1969–2003.
  7. To understand deep learning we need to understand kernel learning. In International Conference on Machine Learning (2018), PMLR, pp. 541–549.
  8. Berezin, F. A. General Concept of Quantization. Communications in Mathematical Physics 40 (1975), 153–174.
  9. Convex optimization. Cambridge University Press, 2004.
  10. Data-driven science and engineering: Machine learning, dynamical systems, and control. Cambridge University Press, 2022.
  11. Kernel analog forecasting: Multiscale test problems. Multiscale Modeling & Simulation 19, 2 (2021), 1011–1040.
  12. Composition operators on the Fock space. Acta Sci. Math.(Szeged) 69, 3-4 (2003), 871–887.
  13. Random matrix theory and symmetric spaces. Physics reports 394, 2-3 (2004), 41–156.
  14. Composition operators on spaces of Entire functions. Proceedings of the American Mathematical Society 135, 7 (2007), 2205–2218.
  15. Deep Neural Tangent Kernel and Laplace Kernel have the same RKHS. arXiv preprint arXiv:2009.10683 (2020).
  16. Colbrook, M. J. The Multiverse of Dynamic Mode Decomposition Algorithms. arXiv preprint arXiv:2312.00137 (2023).
  17. Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems. Communications on Pure and Applied Mathematics 77, 1 (2024), 221–283.
  18. Conway, J. B. A Course in Functional Analysis, vol. 96. Springer, 2019.
  19. On the mean and variance of the generalized inverse of a singular Wishart matrix. Electronic Journal of Statistics 5 (2011).
  20. Cowen, C. C. Composition operators on H2superscript𝐻2H^{2}italic_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. Journal of Operator Theory (1983), 77–106.
  21. Cowen Jr, C. C. Composition operators on spaces of analytic functions. Routledge, 2019.
  22. Koopman spectra in reproducing kernel Hilbert spaces. Applied and Computational Harmonic Analysis 49, 2 (2020), 573–607.
  23. Reproducing kernel Hilbert space compactification of unitary evolution groups. Applied and Computational Harmonic Analysis 54 (2021), 75–136.
  24. Distribution of the generalised inverse of a random matrix and its applications. Journal of Statistical Planning and Inference 136, 1 (2006), 183–192.
  25. Composition operators on Hilbert spaces of entire functions of several variables. Integral Equations and Operator Theory 88 (2017), 301–330.
  26. Random matrix theory. Acta numerica 14 (2005), 233–297.
  27. Fasshauer, G. E. Meshfree approximation methods with MATLAB, vol. 6. World Scientific, 2007.
  28. Dynamic Mode Decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables. Neural Networks 117 (2019), 94–103.
  29. Garnett, J. Bounded Analytic Functions, vol. 236. Springer Science & Business Media, 2007.
  30. On the similarity between the Laplace and Neural Tangent Kernels. Advances in Neural Information Processing Systems 33 (2020), 1451–1461.
  31. On the Similarity between the Laplace and Neural Tangent Kernels. In Advances in Neural Information Processing Systems (2020), H. Larochelle, M. Ranzato, R. Hadsell, M. Balcan, and H. Lin, Eds., vol. 33, Curran Associates, Inc., pp. 1451–1461.
  32. Extraction and prediction of coherent patterns in incompressible flows through space–time Koopman analysis. Physica D: Nonlinear Phenomena 402 (2020), 132211.
  33. On positive functions with positive Fourier transforms. Acta Physica Polonica B 37 (2006), 331.
  34. Modeling Partially Unknown Dynamics with Continuous Time DMD. In 2023 American Control Conference (ACC) (2023), IEEE, pp. 2913–2918.
  35. Boundedness and Compactness of Weighted Composition Operators on Fock Spaces Fp⁢(ℂ)superscript𝐹𝑝ℂ{F}^{p}(\mathbb{C})italic_F start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_C ). Acta Mathematica Vietnamica 41, 3 (2016), 531–537.
  36. Complex symmetric weighted composition operators on the Fock space in several variables. Complex Variables and Elliptic Equations 63, 3 (2018), 391–405.
  37. Weighted Composition Operators on the Mittag-Leffler Spaces of Entire Functions. Complex Analysis and Operator Theory 15, 1 (2021), 1–26.
  38. Hall, B. C. Quantum theory for Mathematicians. Springer, 2013.
  39. Halmos, P. R. A Hilbert space problem book, vol. 19. Springer Science & Business Media, 2012.
  40. On the Rademacher series, Probability in Banach Spaces, Nine, Sandbjerg, Denmark, J. Hoffmann–Jørgensen, J. Kuelbs, MB Marcus, Eds. Birkhauser, Boston 31 (1994), 36.
  41. Kernel machines beat deep neural networks on mask-based single-channel speech enhancement. Proc. Interspeech 2019 (2019), 2748––2752.
  42. Boundedness of composition operators on reproducing kernel Hilbert spaces with analytic positive definite functions. Journal of Mathematical Analysis and Applications 511, 1 (2022), 126048.
  43. Koopman and Perron–Frobenius operators on reproducing kernel Banach spaces. Chaos: An Interdisciplinary Journal of Nonlinear Science 32, 12 (2022).
  44. Ivanov, D. A. Random-matrix ensembles in p-wave vortices. In Vortices in unconventional superconductors and superfluids. Springer, 2002, pp. 253–265.
  45. Neural tangent kernel: Convergence and generalization in neural networks. Advances in Neural Information Processing Systems 31 (2018).
  46. Hankel forms and the Fock space. Revista Matematica Iberoamericana 3, 1 (1987), 61–138.
  47. An Occupation Kernel approach to Optimal Control. arXiv preprint arXiv:2106.00663 (2021).
  48. A kernel-based method for data-driven Koopman spectral analysis. Journal of Computational Dynamics 2, 2 (2016), 247–265.
  49. Khosravi, M. Representer theorem for learning Koopman operators. IEEE Transactions on Automatic Control (2023).
  50. A kernel-based approach to molecular conformation analysis. The Journal of Chemical Physics 149, 24 (2018).
  51. Kernel-based approximation of the Koopman generator and Schrödinger operator. Entropy 22, 7 (2020), 722.
  52. Koopman, B. O. Hamiltonian systems and transformation in Hilbert space. Proceedings of the National Academy of Sciences 17, 5 (1931), 315–318.
  53. On convergence of Extended Dynamic Mode Decomposition to the Koopman operator. Journal of Nonlinear Science 28 (2018), 687–710.
  54. Dynamic Mode Decomposition: Data-Driven Modeling of Complex Dystems. SIAM, 2016.
  55. Multiresolution Dynamic Mode Decomposition. SIAM Journal on Applied Dynamical Systems 15, 2 (2016), 713–735.
  56. Le, T. Normal and isometric weighted composition operators on the Fock space. Bulletin of the London Mathematical Society 46, 4 (2014), 847–856.
  57. Le, T. Composition operators between Segal–Bargmann spaces. Journal of Operator Theory 78, 1 (2017), 135–158.
  58. Lorenz, Edward N. Deterministic nonperiodic flow. Journal of atmospheric sciences 20, 2 (1963), 130–141.
  59. Mehta, M. L. Random matrices. Elsevier, 2004.
  60. Mezić, I. Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dynamics 41 (2005), 309–325.
  61. Mezić, I. Koopman operator, geometry, and learning of dynamical systems. Not. Am. Math. Soc. 68, 7 (2021), 1087–1105.
  62. Comparison of systems with complex behavior. Physica D: Nonlinear Phenomena 197, 1-2 (2004), 101–133.
  63. Dynamic Mode Decomposition of Control-Affine Nonlinear Systems using Discrete Control Liouville Operators. arXiv preprint arXiv:2309.09817 (2023).
  64. Pedersen, G. K. Analysis now, vol. 118. Springer Science & Business Media, 2012.
  65. Peetre, J. The Berezin transform and Ha-plitz operators. Journal of Operator Theory (1990), 165–186.
  66. Error bounds for kernel-based approximations of the Koopman operator. arXiv preprint arXiv:2301.08637 (2023).
  67. Dynamic Mode Decomposition with Control. SIAM Journal on Applied Dynamical Systems 15, 1 (2016), 142–161.
  68. Gaussian Processes for Machine Learning, vol. 1. Springer, 2006.
  69. Reed, M. Methods of modern mathematical physics: Functional Analysis. Elsevier, 2012.
  70. Methods of modern mathematical physics. 1: Functional Analysis. Academic Press, April 1980.
  71. Singular Dynamic Mode Decomposition. SIAM Journal on Applied Dynamical Systems 22, 3 (2023), 2357–2381.
  72. On Occupation Kernels, Liouville operators, and Dynamic Mode Decomposition. In 2021 American Control Conference (ACC) (2021), IEEE, pp. 3957–3962.
  73. Dynamic Mode Decomposition for Continuous Time Systems with the Liouville operator. Journal of Nonlinear Science 32 (2022), 1–30.
  74. Occupation kernels and densely defined Liouville operators for system identification. In 2019 IEEE 58th Conference on Decision and Control (CDC) (2019), IEEE, pp. 6455–6460.
  75. The Mittag Leffler reproducing kernel Hilbert spaces of entire and analytic functions. Journal of Mathematical Analysis and Applications 463, 2 (2018), 576–592.
  76. Theoretical Foundations for the Dynamic Mode Decomposition of High Order Dynamical Systems. arXiv preprint arXiv:2101.02646 (2021).
  77. Spectral analysis of nonlinear flows. Journal of fluid mechanics 641 (2009), 115–127.
  78. Rudin, W. Real and complex analysis. (Mcgraw-Hill International Editions: Mathematics series) (1987).
  79. Rudin, W. Functional Analysis 2nd ed. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York (1991).
  80. Liouville operators over the Hardy space. Journal of Mathematical Analysis and Applications 508, 2 (2022), 125854.
  81. Saitoh, S. Hilbert spaces induced by Hilbert space valued functions. Proceedings of the American Mathematical Society 89, 1 (1983), 74–78.
  82. Saitoh, S. Integral transforms, reproducing kernels and their applications, vol. 369. CRC Press, 1997.
  83. Sarker, I. H. Deep learning: a comprehensive overview on techniques, taxonomy, applications and research directions. SN Computer Science 2, 6 (2021), 420.
  84. Scheffé, H. A useful convergence theorem for probability distributions. The Annals of Mathematical Statistics 18, 3 (1947), 434–438.
  85. Schmid, P. J. Dynamic Mode Decomposition of Numerical and Experimental Data. Journal of fluid mechanics 656 (2010), 5–28.
  86. Schmid, P. J. Application of the Dynamic Mode Decomposition to experimental data. Experiments in fluids 50 (2011), 1123–1130.
  87. Schmid, P. J. Dynamic Mode Decomposition and its variants. Annual Review of Fluid Mechanics 54 (2022), 225–254.
  88. Schölkopf, B. The kernel trick for distances. Advances in Neural Information Processing Systems 13 (2000).
  89. Schröder, E. Ueber iterirte functionen. Mathematische Annalen 3, 2 (1870), 296–322.
  90. Shapiro, J. H. The essential norm of a composition operator. Annals of mathematics (1987), 375–404.
  91. Shapiro, J. H. Composition operators: And Classical Function Theory. Springer Science & Business Media, 2012.
  92. Singh, H. A new kernel function for better AI methods. In 2023 AMS Spring Eastern Sectional Meeting (2023), no. 68-Computer Science, 68T-Artificial Intelligence and 68T07-Artificial Neural Networks and Deep Learning in April 1, 2023, AMS.
  93. Singh, H. An appointment with Reproducing Kernel Hilbert Space generated by Generalized Gaussian RBF as L2−limit-fromsuperscript𝐿2L^{2}-italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT -measure, 2023.
  94. Composition operators on function spaces. Elsevier, 1993.
  95. Complex analysis, vol. 2. Princeton University Press, 2010.
  96. Support vector machines. Springer Science & Business Media, 2008.
  97. An explicit description of the reproducing kernel Hilbert spaces of Gaussian RBF kernels. IEEE Transactions on Information Theory 52, 10 (2006), 4635–4643.
  98. Tao, T. Topics in random matrix theory, vol. 132. American Mathematical Society, 2023.
  99. Fundamental properties and sampling distributions of the multivariate normal distribution. Springer, 1990.
  100. Villani, A. Another note on the inclusion Lp⁢(μ)⊂Lq⁢(μ)superscript𝐿𝑝𝜇superscript𝐿𝑞𝜇L^{p}(\mu)\subset L^{q}(\mu)italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_μ ) ⊂ italic_L start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ( italic_μ ). The American Mathematical Monthly 92, 7 (1985), 485–C76.
  101. Williams, D. Probability with Martingales. Cambridge University Press, 1991.
  102. A data–driven approximation of the Koopman operator: Extending Dynamic Mode Decomposition. Journal of Nonlinear Science 25 (2015), 1307–1346.
  103. A kernel-based method for data-driven Koopman spectral analysis. Journal of Computational Dynamics 2, 2 (2015), 247–265.
  104. Zhao, L. Invertible Weighted Composition Operators on the Fock Space of ℂNsuperscriptℂ𝑁\mathbb{C}^{N}blackboard_C start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT. Journal of Function Spaces 2015 (2015).
  105. Analog forecasting with dynamics-adapted kernels. Nonlinearity 29, 9 (2016), 2888.
  106. Zhu, K. Spaces of Holomorphic Functions in the Unit Ball, vol. 226. Springer, 2005.
  107. Zhu, K. Analysis on Fock Spaces, vol. 263. Springer Science & Business Media, 2012.
  108. Zirnbauer, M. R. Riemannian symmetric superspaces and their origin in random-matrix theory. Journal of Mathematical Physics 37, 10 (1996), 4986–5018.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now