Mori-Zwanzig latent space Koopman closure for nonlinear autoencoder
This addresses the problem of accurately predicting complex nonlinear dynamics in fields like fluid dynamics, offering a novel hybrid method that bridges data-driven techniques with theoretical foundations.
The study tackled the challenge of approximating Koopman operators for nonlinear systems by proposing the Mori-Zwanzig autoencoder (MZ-AE), which integrates a nonlinear autoencoder with a Mori-Zwanzig correction mechanism to achieve closure in latent spaces, resulting in improved predictive capability for flow around a cylinder and promising short-term predictability and robust long-term statistical performance for Kuramoto-Sivashinsky systems.
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.