The Koopman operator presents an attractive approach to achieve global linearization of nonlinear systems, making it a valuable method for simplifying the understanding of complex dynamics. While data-driven methodologies have exhibited promise in approximating finite Koopman operators, they grapple with various challenges, such as the judicious selection of observables, dimensionality reduction, and the ability to predict complex system behaviours accurately. This study presents a novel approach termed Mori-Zwanzig autoencoder (MZ-AE) to robustly approximate the Koopman operator in low-dimensional spaces. The proposed method leverages a nonlinear autoencoder to extract key observables for approximating a finite invariant Koopman subspace and integrates a non-Markovian correction mechanism using the Mori-Zwanzig formalism. Consequently, this approach yields a closed representation of dynamics within the latent manifold of the nonlinear autoencoder, thereby enhancing the precision and stability of the Koopman operator approximation. Demonstrations showcase the technique's ability to capture regime transitions in the flow around a circular cylinder. It also provided a low dimensional approximation for chaotic Kuramoto-Sivashinsky with promising short-term predictability and robust long-term statistical performance. By bridging the gap between data-driven techniques and the mathematical foundations of Koopman theory, MZ-AE offers a promising avenue for improved understanding and prediction of complex nonlinear dynamics.

翻译：Koopman算子为实现非线性系统的全局线性化提供了一种引人注目的途径，使其成为简化复杂动力学理解的宝贵方法。尽管数据驱动方法在近似有限维Koopman算子方面展现出潜力，但其仍面临诸多挑战，例如可观测量选取的审慎性、降维处理以及准确预测复杂系统行为的能力。本研究提出一种名为Mori-Zwanzig自编码器（MZ-AE）的新方法，用于在低维空间中鲁棒地近似Koopman算子。该方法利用非线性自编码器提取关键可观测量以近似有限不变Koopman子空间，并借助Mori-Zwanzig形式引入非马尔可夫修正机制。由此，该方法在非线性自编码器的潜在流形上生成了动力学的闭式表示，从而提升了Koopman算子近似的精度与稳定性。示例展示了该技术捕捉圆柱绕流中流态跃迁的能力，同时为混沌Kuramoto-Sivashinsky方程提供了低维近似，具有出色的短期可预测性和稳健的长期统计性能。通过弥合数据驱动技术与Koopman理论数学基础之间的鸿沟，MZ-AE为改进复杂非线性动力学的理解与预测提供了一条有前景的路径。