Temporally Consistent Koopman Autoencoders for Forecasting Dynamical Systems

TL;DR

This paper introduces tcKAE, a novel Koopman Autoencoder variant that improves long-term forecasting of high-dimensional dynamical systems by enforcing temporal consistency, especially effective with limited and noisy data.

Contribution

The paper proposes a temporally consistent regularization for Koopman Autoencoders, enhancing their robustness and predictive accuracy in data-scarce, noisy environments.

Findings

tcKAE outperforms existing KAEs in various test cases.

The regularization improves long-term prediction accuracy.

The approach is theoretically justified by Koopman spectral theory.

Abstract

Absence of sufficiently high-quality data often poses a key challenge in data-driven modeling of high-dimensional spatio-temporal dynamical systems. Koopman Autoencoders (KAEs) harness the expressivity of deep neural networks (DNNs), the dimension reduction capabilities of autoencoders, and the spectral properties of the Koopman operator to learn a reduced-order feature space with simpler, linear dynamics. However, the effectiveness of KAEs is hindered by limited and noisy training datasets, leading to poor generalizability. To address this, we introduce the temporally consistent Koopman autoencoder (tcKAE), designed to generate accurate long-term predictions even with limited and noisy training data. This is achieved through a consistency regularization term that enforces prediction coherence across different time steps, thus enhancing the robustness and generalizability of tcKAE over existing models. We provide analytical justification for this approach based on Koopman spectral theory and empirically demonstrate tcKAE's superior performance over state-of-the-art KAE models across a variety of test cases, including simple pendulum oscillations, kinetic plasma, and fluid flow data.