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Data-driven discovery with Limited Data Acquisition for fluid flow across cylinder

Himanshu Singh

TL;DR

This work tackles the challenge of discovering governing principles for dynamical systems when data are limited, focusing on fluid flow across a cylinder. It develops a data-driven framework based on Kernelized Extended Dynamic Mode Decomposition (KeDMD) within a Reproducing Kernel Hilbert Space (RKHS) generated by the normalized Laplacian measure $d\mu_{\sigma,1,\mathbb{C}^n}$, and demonstrates via random-matrix augmentation that dominant Koopman modes can be recovered with very few snapshots. The authors prove key operator-theoretic properties for Koopman operators on this RKHS, including boundedness, essential-norm estimates, compactness, and, notably, closability, which is shown to hold for the Laplacian-measure RKHS but not for the Gaussian RBF RKHS. Experimental results on cylinder wake data show that the Laplacian kernel outperforms GRBF in limited-data regimes, yielding real and imaginary Koopman modes and informative Gram matrices, while GRBF requires more data to achieve similar fidelity. Overall, the paper provides a rigorous operator-theoretic foundation and practical algorithm for data-efficient, kernel-based discovery of dynamical principals in fluid flows, with implications for robust model reduction under data scarcity.

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 $dμ_{σ,1,\mathbb{C}^n}(z)=(2πσ^2)^{-n}\exp\left(-\frac{\|z\|_2}σ\right)dV(z)$ when it is embedded in $L^2-$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 $dμ_{σ,1,\mathbb{C}^n}(z)$ in the $L^2-$sense. In doing so, we provide the compactification & closable characterization of Koopman operator over the RKHS generated by the normalized Laplacian measure in the $L^2-$sense.

Data-driven discovery with Limited Data Acquisition for fluid flow across cylinder

TL;DR

This work tackles the challenge of discovering governing principles for dynamical systems when data are limited, focusing on fluid flow across a cylinder. It develops a data-driven framework based on Kernelized Extended Dynamic Mode Decomposition (KeDMD) within a Reproducing Kernel Hilbert Space (RKHS) generated by the normalized Laplacian measure , and demonstrates via random-matrix augmentation that dominant Koopman modes can be recovered with very few snapshots. The authors prove key operator-theoretic properties for Koopman operators on this RKHS, including boundedness, essential-norm estimates, compactness, and, notably, closability, which is shown to hold for the Laplacian-measure RKHS but not for the Gaussian RBF RKHS. Experimental results on cylinder wake data show that the Laplacian kernel outperforms GRBF in limited-data regimes, yielding real and imaginary Koopman modes and informative Gram matrices, while GRBF requires more data to achieve similar fidelity. Overall, the paper provides a rigorous operator-theoretic foundation and practical algorithm for data-efficient, kernel-based discovery of dynamical principals in fluid flows, with implications for robust model reduction under data scarcity.

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 when it is embedded in sense for the holomorphic functions over , 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 in the sense. In doing so, we provide the compactification & closable characterization of Koopman operator over the RKHS generated by the normalized Laplacian measure in the sense.
Paper Structure (50 sections, 32 theorems, 133 equations, 16 figures, 1 table)

This paper contains 50 sections, 32 theorems, 133 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Dynamical Systems & Hilbert space
  3. Interface between Operator theory & Dynamical System
  4. What this paper offers?
  5. Motivation for this paper
  6. Offerings of this paper
  7. Detail plan of the paper
  8. DMD with Limited Data Acquisition
  9. Extended Dynamic Mode Decomposition
  10. Review of eDMD
  11. On Limited Data Acquisition
  12. Using random matrix theory for limited data acquisition
  13. Preliminary to the Random Matrix Theory
  14. Choice of random matrix
  15. Theory for augmenting data-set snapshots with random matrix
  16. ...and 35 more sections

Key Result

Theorem 1.1

Let $H$ be an RKHS over an nonempty set $X$, Then $k:X\times X\to\mathbf{K}$ defined as $k(x,x')\coloneqq \langle\mathcal{E}_x,\mathcal{E}_{x'}\rangle_H$ for $x,x'\in X$ is the only reproducing kernel of $H$. Additionally, for some index set $\mathcal{I}$, if we have $\left\{\mathbf{e}_i\right\}_{i\ with an absolute convergence.

Figures (16)

  • Figure 1: L63 dynamical system
  • Figure 2: NN Simulation of L63 in $x(t),~y(t)$ and $z(t)-$coordinates
  • Figure 3: A conceptual diagram to intercept the different notions of data acquisition. The very-first matrix in the extreme left symbolised the ideal or at-least expected condition of having-supposedly $m-$snapshots. The unfortunate situation of acquiring only $m_0-$snapshots out of those $m-$snapshots is depicted in the middle figure. In order to construct the ideal situation matrix in which $m-$snapshots are present, a simple padding of $m-m_0$ random vectors is inserted and this is symbolized in extreme right.
  • Figure 4: Interactive figure for inter-connection of various exponential type measures as depicted in \ref{['table-1exponentialtype']}.
  • Figure 5: Vector field: Affine dynamic (L) & Non-affine dynamic (R).
  • ...and 11 more figures

Theorems & Definitions (73)

  • Definition 1: Kernel Function
  • Example 1
  • Definition 2: Reproducing Kernel Hilbert Space
  • Theorem 1.1: Moore-Aronszajn Theorem aronszajn1950theory
  • Example 2
  • Lemma 1: Lemma 2.4 in le2017composition
  • Definition 3: Koopman Operators
  • Definition 4: Liouville Operator
  • Definition 5
  • Definition 6
  • ...and 63 more