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.

MZ-AE 方法使用非线性自编码器来提取关键观测量以在低维空间中近似 Koopman 算符，并利用 Mori-Zwanzig 形式主义的非马尔可夫校正机制，将系统的动态模型映射到非线性自编码器的潜在空间，从而提高了 Koopman 算符近似的精度和稳定性，架起了数据驱动技术与 Koopman 理论之间的桥梁，为复杂非线性动力学的理解和预测提供了一种有前途的途径。