Multiscale Modelling with Physics-informed Neural Network: from Large-scale Dynamics to Small-scale Predictions in Complex Systems

Multiscale phenomena manifest across various scientific domains, presenting a ubiquitous challenge in accurately and effectively predicting multiscale dynamics in complex systems. In this paper, a novel decoupling solving mode 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. 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. This paper presents a valuable and promising approach to enhance the computational simulations of multiscale spatiotemporal systems, which enables the acquisition of large-scale data with minimal computational demands, followed by Spectral PINN to capture small-scale dynamics with improved efficiency and accuracy.