Modeling of high-dimensional time-delay chaotic system based on Fourier neural operator

Dynamical systems with time-delays are prevalent in diverse scientific disciplines as high-dimensional chaotic systems. These systems are described by delay differential equations and have been extensively studied. Since the dynamical equations of most existing time-delay chaotic systems are unknown, the modeling of the time-delay chaotic system is of significant scientific and technological importance. In this work, we first investigate the high-dimensional chaotic properties of the time-delay chaotic system with a constant delay, and then propose a deep learning modeling approach for time-delay chaotic systems based on its inherent historical state dependence. Subsequently, we validate our scheme for two typical time-delay chaotic systems: the Mackey-Glass (M-G) system and the Ikeda system. In particular, we introduce a deep learning architecture with infinite-dimensional function space mapping capabilities called the Fourier Neural Operator (FNO) for modeling the time-delay chaotic system, and compare its modeling performance with that of traditional feed-forward neural networks (FNN) and long short-term memory (LSTM) network. The results demonstrate that FNO exhibits outstanding modeling performance, achieving an R-squared (R2) coefficient of around 0.99 under small amount of data conditions, whereas the R2 coefficients for FNN and LSTM are below 0.95. The R2 performance results indicate that the FNO can accurately reproduce the dynamics of chaotic systems up to approximately 8 times the Lyapunov time. In addition, we further validate two widely used optical time-delay chaotic systems, and the results show that our approach exhibits excellent modeling performance in practical optical chaotic systems as well.