Dynamical systems provide a comprehensive way to study complex and changing behaviors across various sciences. Many modern systems are too complicated to analyze directly or we do not have access to models, driving significant interest in learning methods. Koopman operators have emerged as a dominant approach because they allow the study of nonlinear dynamics using linear techniques by solving an infinite-dimensional spectral problem. However, current algorithms face challenges such as lack of convergence, hindering practical progress. This paper addresses a fundamental open question: \textit{When can we robustly learn the spectral properties of Koopman operators from trajectory data of dynamical systems, and when can we not?} Understanding these boundaries is crucial for analysis, applications, and designing algorithms. We establish a foundational approach that combines computational analysis and ergodic theory, revealing the first fundamental barriers -- universal for any algorithm -- associated with system geometry and complexity, regardless of data quality and quantity. For instance, we demonstrate well-behaved smooth dynamical systems on tori where non-trivial eigenfunctions of the Koopman operator cannot be determined by any sequence of (even randomized) algorithms, even with unlimited training data. Additionally, we identify when learning is possible and introduce optimal algorithms with verification that overcome issues in standard methods. These results pave the way for a sharp classification theory of data-driven dynamical systems based on how many limits are needed to solve a problem. These limits characterize all previous methods, presenting a unified view. Our framework systematically determines when and how Koopman spectral properties can be learned.

翻译：动力学系统为研究各学科中复杂且变化的行为提供了全面的方法。许多现代系统过于复杂而无法直接分析，或者我们无法获得其模型，这推动了对学习方法的广泛关注。Koopman算子已成为一种主导方法，因为它通过求解无限维谱问题，允许使用线性技术研究非线性动力学。然而，当前算法面临缺乏收敛性等挑战，阻碍了实际进展。本文解决了一个基本的开放性问题：\textit{我们何时能够从动力学系统的轨迹数据中稳健地学习Koopman算子的谱特性，何时不能？}理解这些边界对于分析、应用和算法设计至关重要。我们建立了一种结合计算分析和遍历理论的基础性方法，揭示了与系统几何和复杂性相关的首个基本障碍——这些障碍对任何算法都具有普适性，且与数据质量和数量无关。例如，我们证明了环面上表现良好的光滑动力学系统中，Koopman算子的非平凡特征函数无法通过任何（即使是随机化的）算法序列确定，即使训练数据无限。此外，我们确定了学习可行的条件，并引入了带有验证机制的最优算法，这些算法克服了标准方法中的问题。这些结果为基于解决问题所需极限数量的数据驱动动力学系统尖锐分类理论铺平了道路。这些极限刻画了所有现有方法，提供了统一视角。我们的框架系统性地确定了Koopman谱特性何时及如何能够被学习。