Scholarlink

<p>Koopman theory provides an approach to lift nonlinear dynamical systems into a linear measurement function space governed by the Koopman operator. Unfortunately, the lifted systems often encounter high-dimensional or even infinite-dimensional challenges due to complex nonlinear structures. Effective approximation of the Koopman operator from state-space measurements is crucial. This paper proposes a Fourier kernel-based method (FKM), which provides a universal approximation framework for the Koopman operator in reproducing kernel Hilbert space (RKHS). The action of this linear operator in RKHS is represented through Fourier kernel integration. Depending on observed state data, this integration can be reduced from the entire state space to the dynamic range captured by the data, significantly improving computational efficiency. An additional advantage stems from sparse sampling of Fourier modes and integration points, which follow complex distributions rather than uniform distributions. These parameters can be further optimized using deep learning techniques. The method’s effectiveness is demonstrated through two case studies: a 1D discrete system and a 2D continuous system. Results show that FKM performs well even with small-sample datasets.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Fourier kernel-based data driven approximation of the Koopman operator

  • 期刊论文

  • 发表日期：

    2025年6月17日

查看全文
  • Peipei Liang,
  • Jinqian Feng,
  • Jin Su,
  • Qin Guo,
  • Youpan Han

摘要

Koopman theory provides an approach to lift nonlinear dynamical systems into a linear measurement function space governed by the Koopman operator. Unfortunately, the lifted systems often encounter high-dimensional or even infinite-dimensional challenges due to complex nonlinear structures. Effective approximation of the Koopman operator from state-space measurements is crucial. This paper proposes a Fourier kernel-based method (FKM), which provides a universal approximation framework for the Koopman operator in reproducing kernel Hilbert space (RKHS). The action of this linear operator in RKHS is represented through Fourier kernel integration. Depending on observed state data, this integration can be reduced from the entire state space to the dynamic range captured by the data, significantly improving computational efficiency. An additional advantage stems from sparse sampling of Fourier modes and integration points, which follow complex distributions rather than uniform distributions. These parameters can be further optimized using deep learning techniques. The method’s effectiveness is demonstrated through two case studies: a 1D discrete system and a 2D continuous system. Results show that FKM performs well even with small-sample datasets.