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 solving mode is proposed
for characterizing multiscale dynamics through a decoupling method. By
modelling large-scale dynamics independently and treating small-scale dynamics
as a slaved system, a Spectral PINN is developed to approach the small-scale
system in an orthogonal basis functional space. The effectiveness of the method
is demonstrated through extensive numerical experiments, including
one-dimensional Kuramot-Sivashinsky (KS) equation, two- and three-dimensional
Navier-Stokes (NS) 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
novel decoupling approach simplifies the analysis and prediction of
spatiotemporal systems, where large-scale data can be obtained with low
computational demands, followed by Spectral PINNs for capturing small-scale
dynamics with improved efficiency and accuracy.

通过解耦方法，本文提出了一种用于表征多尺度动力学的新求解模式，该模式通过将大尺度动力学独立建模并将小尺度动力学视为其受控系统，在正交基函数空间中开发了一种谱 PINN 来逼近小尺度系统。该方法的有效性通过广泛的数值实验得到了证明，包括一维 Kuramot-Sivashinsky (KS) 方程，二维和三维 Navier-Stokes (NS) 方程，展示了其在解决流体动力学问题方面的多功能性。此外，我们还深入研究了该方法在更复杂情况下的应用，包括非均匀网格，复杂几何，带有噪声的大尺度数据和高维小尺度动力学的问题。对于这些情景的讨论有助于全面了解该方法的能力和局限性。这种新的解耦方法简化了时空系统的分析和预测，其中可以通过低计算要求获取大尺度数据，然后通过谱 PINNs 捕捉到具有改进的效率和准确性的小尺度动力学。