用梯度优化自动学习动态系统基函数，无需先验知识
Learning dynamical systems from data: Gradient-based dictionary optimization
- 基于梯度下降自动优化基函数，替代人工选取
- 在多个经典系统上实现高精度建模，误差显著降低
- 适合无领域知识的科研人员快速构建动力系统模型
Koopman算子在分析动态系统全局行为中起关键作用。现有数据驱动方法通常依赖固定的基函数（即字典），而最优基函数的选择高度依赖具体问题且常需领域知识。本文提出一种基于梯度下降的字典优化框架，可从数据中学习合适且可解释的基函数，并可与EDMD、SINDy和PDE-FIND结合使用。通过奥恩斯坦-乌伦贝克过程、楚亚电路、非线性热方程以及蛋白质折叠数据等基准问题验证了该方法的有效性。
原文摘要 · Abstract (English)
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.
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