一种求解多尺度问题的新范式
A Novel Paradigm in Solving Multiscale Problems
AI总结：
本文提出一种解耦求解多尺度问题的新范式，独立建模大尺度动力学并将小尺度动力学作为从属系统，利用谱物理信息神经网络高效刻画小尺度动力学，并通过多种流体动力学数值实验验证其有效性与适用性。
摘要：
多尺度现象广泛存在于各个科学领域中，对复杂系统中的多尺度动力学进行准确而有效的模拟一直是一个普遍存在的挑战。本文提出了一种新颖的解耦求解范式，即独立建模大尺度动力学，并将小尺度动力学视为一个受约束的从属系统。我们开发了一种谱物理信息神经网络（Spectral PINN），用于高效且准确地刻画小尺度系统，从而应对神经网络中多尺度动力学表示所带来的挑战。通过大量数值实验验证了该方法的有效性，包括一维Kuramoto-Sivashinsky方程、二维和三维Navier-Stokes方程，展示了其在处理流体动力学问题上的广泛适用性。此外，我们还深入探讨了所提方法在更复杂问题中的应用，包括非均匀网格、复杂几何形状、含噪声的大尺度数据以及高维小尺度动力学。针对这些场景的讨论有助于全面理解该方法的能力与局限性。通过以极低的计算代价获取大尺度数据，并结合Spectral PINN对小尺度动力学进行高效准确的刻画，我们的方法为寻求有效应对多尺度现象的研究人员提供了一种有价值且有前景的途径。
英文摘要：
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.
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