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 behaviors 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 cylinder. It also provides a low dimensional approximation for 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算子为非线性系统提供了实现全局线性化的有效途径，成为简化复杂动力系统理解的重要方法。虽然数据驱动方法在近似有限维Koopman算子方面展现出潜力，但仍面临观测变量选择、降维及复杂系统行为准确预测等多重挑战。本研究提出一种名为Mori-Zwanzig自编码器（MZ-AE）的新方法，用于在低维空间中稳健近似Koopman算子。该方法利用非线性自编码器提取关键观测变量以近似有限不变Koopman子空间，并借助Mori-Zwanzig形式化框架集成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实现了动力系统的闭合表示，从而增强了Koopman算子近似的精度与稳定性。数值实验表明，该技术能够捕捉圆柱绕流中的模态转变，同时为Kuramoto-Sivashinsky方程提供低维近似，兼具优异的短期预测能力与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为理解和预测复杂非线性动力学开辟了新的前景。