Mori-Zwanzig latent space Koopman closure for nonlinear autoencoder

TL;DR

This paper introduces Mori-Zwanzig autoencoder (MZ-AE), a novel method that combines nonlinear autoencoders with Mori-Zwanzig formalism to improve Koopman operator approximation for complex nonlinear systems, enhancing prediction accuracy and stability.

Contribution

The paper presents a new approach integrating autoencoders and Mori-Zwanzig formalism to robustly approximate Koopman operators in low-dimensional latent spaces.

Findings

Improved predictive accuracy for flow around a cylinder.

Effective low-dimensional approximation for Kuramoto-Sivashinsky system.

Enhanced long-term statistical stability in predictions.

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.