利用子系统方程先验信息，提升耦合系统建模的数据效率
Learning Koopman operators for coupled systems via information on governing equations of subsystems
- 基于子系统微分方程构建柯尔曼算子近似，引入先验知识
- 仅用少量快照即可准确学习耦合效应，提升数据效率
- 适合高维耦合系统建模，尤其适用于数据稀缺场景
非线性耦合系统在科学与工程中普遍存在。由于其高维性及子系统间复杂相互作用，分析与建模极具挑战。近年来，基于柯尔曼算子的算子理论方法成为分析非线性动力系统的有力工具。扩展动态模态分解（EDMD）是最流行的柯尔曼算子近似方法之一。然而，EDMD是纯数据驱动方法，在数据有限时可能不稳定且不准确。本文提出一种新方法，利用各子系统的控制微分方程构造耦合系统的有限维柯尔曼算子近似。该方法通过已知子系统动力学作为先验信息，从少量快照中学习耦合引起的修正项，从而提升数据效率。我们通过耦合振子系统的数值实验验证了该方法的有效性。
原文摘要 · Abstract (English)
Nonlinear coupled systems are ubiquitous in science and engineering. The analysis and modeling of such systems are challenging due to their high dimensionality and complex interactions among subsystems. In recent years, operator-theoretic methods based on the Koopman operator have attracted attention as a powerful tool for analyzing and modeling nonlinear dynamical systems. Extended dynamic mode decomposition (EDMD) is one of the most popular methods for approximating the Koopman operator. However, EDMD is a purely data-driven method, and it may be unstable and inaccurate for coupled systems under limited data availability. In this paper, we propose a method to construct a finite-dimensional Koopman approximation for coupled systems using the differential equations governing each subsystem. The proposed method aims to improve data efficiency by using the known subsystem dynamics as prior information and learning the coupling-induced correction from a limited number of snapshots. We also demonstrate its effectiveness through numerical experiments on coupled oscillator systems.
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