In Koopman operator theory, a finite-dimensional nonlinear system is transformed into an infinite but linear system using a set of observable functions. However, manually selecting observable functions that span the invariant subspace of the Koopman operator based on prior knowledge is inefficient and challenging, particularly when little or no information is available about the underlying systems. Furthermore, current methodologies tend to disregard the importance of the invertibility of observable functions, which leads to inaccurate results. To address these challenges, we propose the so-called FlowDMD, a Flow-based Dynamic Mode Decomposition that utilizes the Coupling Flow Invertible Neural Network (CF-INN) framework. FlowDMD leverages the intrinsically invertible characteristics of the CF-INN to learn the invariant subspaces of the Koopman operator and accurately reconstruct state variables. Numerical experiments demonstrate the superior performance of our algorithm compared to state-of-the-art methodologies.

翻译：在Koopman算子理论中，有限维非线性系统可通过一组观测函数转化为无限维但线性的系统。然而，基于先验知识手动选取能够张成Koopman算子不变子空间的观测函数既低效又困难，尤其是在对底层系统信息知之甚少或完全未知的情况下。此外，现有方法往往忽视观测函数可逆性的重要性，导致结果不够精确。为解决上述挑战，我们提出FlowDMD（基于流的动态模式分解），该算法利用耦合流可逆神经网络（CF-INN）框架。FlowDMD借助CF-INN固有的可逆特性，学习Koopman算子的不变子空间并精确重构状态变量。数值实验结果表明，与现有最优方法相比，本算法具有更优越的性能。