  Multiscale phenomena manifest across various scientific domains, presenting a
ubiquitous challenge in accurately and effectively simulating multiscale
dynamics in complex systems. In this paper, a novel decoupling solving paradigm
is proposed through modelling large-scale dynamics independently and treating
small-scale dynamics as a slaved system. A Spectral Physics-informed Neural
Network (PINN) is developed to characterize the small-scale system in an
efficient and accurate way, addressing the challenges posed by the
representation of multiscale dynamics in neural networks. The effectiveness of
the method is demonstrated through extensive numerical experiments, including
one-dimensional Kuramot-Sivashinsky equation, two- and three-dimensional
Navier-Stokes equations, showcasing its versatility in addressing problems of
fluid dynamics. Furthermore, we also delve into the application of the proposed
approach to more complex problems, including non-uniform meshes, complex
geometries, large-scale data with noise, and high-dimensional small-scale
dynamics. The discussions about these scenarios contribute to a comprehensive
understanding of the method's capabilities and limitations. By enabling the
acquisition of large-scale data with minimal computational demands, coupled
with the efficient and accurate characterization of small-scale dynamics via
Spectral PINN, our approach offers a valuable and promising approach for
researchers seeking to tackle multiscale phenomena effectively.
