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, aka 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，有限维非线性系统通过一组可观测量函数被转化为无限维但线性的系统。然而，基于先验知识手动选择能够张成Koopman算子不变子空间的可观测量函数既低效又困难，特别是在对底层系统了解甚少或完全未知的情况下。此外，现有方法往往忽视可观测量函数可逆性的重要程度，导致结果不精确。为解决这些问题，我们提出名为FlowDMD（基于流的动态模态分解）的方法，该方法利用耦合流可逆神经网络（CF-INN）框架。FlowDMD借助CF-INN固有的可逆特性来学习Koopman算子的不变子空间，并精确重构状态变量。数值实验表明，与现有最先进方法相比，我们的算法展现出优越性能。