The Koopman operator plays a crucial role in analyzing the global behavior of dynamical systems. Existing data-driven methods for approximating the Koopman operator or discovering the governing equations of the underlying system typically require a fixed set of basis functions, also called dictionary. The optimal choice of basis functions is highly problem-dependent and often requires domain knowledge. We present a novel gradient descent-based optimization framework for learning suitable and interpretable basis functions from data and show how it can be used in combination with EDMD, SINDy, and PDE-FIND. We illustrate the efficacy of the proposed approach with the aid of various benchmark problems such as the Ornstein-Uhlenbeck process, Chua's circuit, a nonlinear heat equation, as well as protein-folding data.

翻译：Koopman算子在分析动力系统的全局行为中起着关键作用。现有用于逼近Koopman算子或发现底层系统控制方程的基于数据驱动的方法通常需要一组固定的基函数（也称为字典）。基函数的最优选择高度依赖于具体问题，且往往需要领域知识。我们提出了一种新颖的基于梯度下降的优化框架，用于从数据中学习合适且可解释的基函数，并展示了如何将其与EDMD、SINDy和PDE-FIND等方法结合使用。我们借助各种基准问题（如Ornstein-Uhlenbeck过程、蔡氏电路、非线性热方程以及蛋白质折叠数据）说明了所提方法的有效性。