🤖 AI Summary
Accurately and stably approximating the Koopman operator for high-dimensional nonlinear systems remains challenging in data-driven settings. To address this, we propose the Mori–Zwanzig Autoencoder (MZ-AE), the first framework to embed the Mori–Zwanzig non-Markovian projection formalism into a nonlinear autoencoding architecture. MZ-AE constructs a memory-corrected Koopman operator closure in a low-dimensional latent space, enabling robust approximation of the invariant Koopman subspace and dynamical closure on the underlying manifold. Compared with conventional Koopman learning methods, MZ-AE achieves significantly improved short-term prediction accuracy on canonical PDE-based systems—including cylinder wake flow and the Kuramoto–Sivashinsky equation—while preserving long-term statistical stability. By unifying physical interpretability with data adaptivity, MZ-AE establishes a novel paradigm for modeling nonlinear dynamical systems.
📝 Abstract
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 an approximate closure of the dynamics within the latent manifold of the nonlinear autoencoder, thereby enhancing the accuracy and stability of the Koopman operator approximation. Demonstrations showcase the technique's improved predictive capability for flow around a cylinder. It also provides a low dimensional approximation for Kuramoto-Sivashinsky (KS) 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.
