Mori-Zwanzig latent space Koopman closure for nonlinear autoencoder

TL;DR

This work tackles the challenge of obtaining a finite, low-dimensional Koopman representation for nonlinear dynamics by marrying a nonlinear autoencoder with Mori–Zwanzig memory to close the reduced model. The proposed Mori-Zwanzig autoencoder (MZ-AE) learns resolved observables in a latent space and applies a non-Markovian closure (via a memory kernel or RNN) to form a generalized Langevin equation that governs latent dynamics. The approach yields substantial improvements over standard Koopman-operator methods and DMD in predicting flow around a cylinder and in the KS system, achieving accurate short-term forecasts and robust long-term statistics with few observables. This latent-space closure enhances predictive stability and offers a principled, interpretable ROM for complex nonlinear dynamics with potential broad applicability in fluids and other high-dimensional 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.

Figures (11)

• Figure 1: Schematic for the MZ-AE framework.
• Figure 2: Top: Computational domain and sampled sub-domain (dotted rectangle) for cylinder flow. Bottom: Data points sampled from the sub-domain for training.
• Figure 3: Comparative study across different models in the limit cycle regime of the cylinder flow (Case I). Top-left: RMSE of the predicted trajectories with respect to the true solution (direct numerical simulation (DNS)) with increasing number of observables ($r$). Top-right: kinetic energy predictions in state space across different models with $r=2$ resolved observables. Bottom: long-term kinetic energy trajectories comprising 1000 timeunits (corresponding to 180 vortex shedding cycles) of state variables. The x-axis represents normalized time $t^*=n\Delta tD/U_{\infty}$.
• Figure 4: Relative mean squared prediction errors across different numbers of resolved observables ($r$) and the memory length ($q$) in the limit cycle of cylinder flow (Case I). Left: MZ-AE GFDc, Right: MZi-AE LRNN.
• Figure 5: Comparative study across different models from the baseflow to the limit cycle (Case II) using $r=2$ resolved observables. Left: For MZi-AE NRNN, $L^2$ norm of the contributions to the observable dynamics from the MZ-AE Markov operator ($\mathbf{\tilde{K}}_{\Delta}\hat{\mathbf{g}}$) and the memory model ($\bm{\xi}$). Right: kinetic energy of predicted trajectories in the state space across different models and the true solution (DNS). The x-axis represents normalized time $t^*=n\Delta tD/U_{\infty}$.