Koopman dynamic-oriented deep learning for invariant subspace identification and full-state prediction of complex systems

Abstract

One strategy for predicting the state of nonlinear dynamical systems (typically of high dimensionality) is global linearization, such as utilizing the Koopman analysis model to transform the system state into an invariant subspace that evolves linearly. A critical challenge in the Koopman model is designing or deriving observation functions, typically nonlinear, to linearize the dynamical systems. To address the challenge, this study proposes a model called SKCAE (Skip-connected Koopman Convolutional AutoEncoder) that combines guidance of physics and data to learn observation functions accurately and efficiently. The novelties of SKCAE are twofold: (1) a coordinate-transforming main network for the construction of nonlinear observation functions, enabling the explicit identification of the low-dimensional dynamics that dominate the system's dynamical evolution, and (2) a Koopman dynamics-oriented subnetwork for quantitatively interpreting the system's intrinsic mechanisms from the frequency perspective and efficiently predicting the system state. Four numerical cases with discrete/continuous spectra on Eulerian/Lagrangrian descriptions are studied, i.e., fixed-point attractor, Duffing oscillator, fluid flow past a cylinder, and channel turbulence. Results demonstrate that SKCAE achieves significant improvement (up to a hundred times) in accuracy compared to conventional models and possesses remarkable capability in handling scenarios with data noises and/or loss, owing to its intrinsic advantage in retaining a broad range of frequency components of a dynamical system.