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The Koopman operator presents an attractive approach to achieve globallinearization of nonlinear systems, making it a valuable method for simplifyingthe understanding of complex dynamics. While data-driven methodologies haveexhibited promise in approximating finite Koopman operators, they grapple withvarious challenges, such as the judicious selection of observables,dimensionality reduction, and the ability to predict complex system behavioursaccurately. This study presents a novel approach termed Mori-Zwanzigautoencoder (MZ-AE) to robustly approximate the Koopman operator inlow-dimensional spaces. The proposed method leverages a nonlinear autoencoderto extract key observables for approximating a finite invariant Koopmansubspace and integrates a non-Markovian correction mechanism using theMori-Zwanzig formalism. Consequently, this approach yields a closedrepresentation of dynamics within the latent manifold of the nonlinearautoencoder, thereby enhancing the precision and stability of the Koopmanoperator approximation. Demonstrations showcase the technique's ability tocapture regime transitions in the flow around a circular cylinder. It alsoprovided a low dimensional approximation for chaotic Kuramoto-Sivashinsky withpromising short-term predictability and robust long-term statisticalperformance. By bridging the gap between data-driven techniques and themathematical foundations of Koopman theory, MZ-AE offers a promising avenue forimproved understanding and prediction of complex nonlinear dynamics.

Mori-Zwanzig latent space Koopman closure for nonlinear autoencoder