One strategy for predicting the state of nonlinear dynamical systems (typically of high dimensionality) is global linearization, such as utilizing the Koopman analysis model to transform the system state into an invariant subspace that evolves linearly. A critical challenge in the Koopman model is designing or deriving observation functions, typically nonlinear, to linearize the dynamical systems. To address the challenge, this study proposes a model called SKCAE (Skip-connected Koopman Convolutional AutoEncoder) that combines guidance of physics and data to learn observation functions accurately and efficiently. The novelties of SKCAE are twofold: (1) a coordinate-transforming main network for the construction of nonlinear observation functions, enabling the explicit identification of the low-dimensional dynamics that dominate the system's dynamical evolution, and (2) a Koopman dynamics-oriented subnetwork for quantitatively interpreting the system's intrinsic mechanisms from the frequency perspective and efficiently predicting the system state. Four numerical cases with discrete/continuous spectra on Eulerian/Lagrangrian descriptions are studied, i.e., fixed-point attractor, Duffing oscillator, fluid flow past a cylinder, and channel turbulence. Results demonstrate that SKCAE achieves significant improvement (up to a hundred times) in accuracy compared to conventional models and possesses remarkable capability in handling scenarios with data noises and/or loss, owing to its intrinsic advantage in retaining a broad range of frequency components of a dynamical system.

中文翻译：

---

库普曼动态深度学习用于复杂系统的不变子空间识别和全状态预测


预测非线性动力系统（通常是高维）状态的一种策略是全局线性化，例如利用库普曼分析模型将系统状态转换为线性演化的不变子空间。库普曼模型中的一个关键挑战是设计或导出观测函数（通常是非线性的）以使动力系统线性化。为了应对这一挑战，本研究提出了一种名为 SKCAE（跳跃连接库普曼卷积自动编码器）的模型，该模型结合了物理和数据的指导来准确有效地学习观测函数。 SKCAE 的新颖性有两个：(1) 用于构建非线性观测函数的坐标变换主网络，能够显式识别主导系统动态演化的低维动力学；(2) 面向库普曼动力学的模型用于从频率角度定量解释系统内在机制并有效预测系统状态的子网络。研究了欧拉/拉格朗日描述上具有离散/连续谱的四种数值情况，即定点吸引子、杜芬振荡器、经过圆柱体的流体流动和通道湍流。结果表明，SKCAE 与传统模型相比，在精度上取得了显着提高（高达一百倍），并且由于其在保留广泛的频率分量方面的内在优势，在处理数据噪声和/或丢失的场景方面具有出色的能力。动力系统。