用托普利茨矩阵方法从数据中提取动力系统的谱特性
Toeplitz Based Spectral Methods for Data-driven Dynamical Systems
- 基于托普利茨滤波器估计系统演化算子的谱
- 可准确恢复确定性和混沌系统中的特征值与特征函数
- 适合研究无方程动力系统的数据驱动建模者
我们提出一种基于托普利茨的框架，用于从数据中驱动估计动力系统中线性演化算子的谱。针对无法获取运动方程的平衡轨迹，该方法对无穷小生成元应用托普利茨滤波器，以提取特征值、特征函数和谱测度。可通过设计融入自伴随或斜对称等结构先验知识。该方法在统计上一致且计算高效，利用了统计学习中常见的原始与对偶算法。数值实验表明，该框架可恢复标准数据驱动方法无法达到的谱特性。
原文摘要 · Abstract (English)
We introduce a Toeplitz-based framework for data-driven spectral estimation of linear evolution operators in dynamical systems. Focusing on transfer and Koopman operators from equilibrium trajectories without access to the underlying equations of motion, our method applies Toeplitz filters to the infinitesimal generator to extract eigenvalues, eigenfunctions, and spectral measures. Structural prior knowledge, such as self-adjointness or skew-symmetry, can be incorporated by design. The approach is statistically consistent and computationally efficient, leveraging both primal and dual algorithms commonly used in statistical learning. Numerical experiments on deterministic and chaotic systems demonstrate that the framework can recover spectral properties beyond the reach of standard data-driven methods.
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