标题： 物理信息神经网络中的多级数据集训练方法 显示英文标题

标题： Multi-level datasets training method in Physics-Informed Neural Networks

摘要： 物理信息神经网络（PINNs）作为一种有前途的方法，在解决偏微分方程（PDEs）方面崭露头角，并在计算机科学和各种与物理相关的领域引起了广泛关注。尽管已经证明了它们能够整合多种应用中的物理定律，但PINNs仍然难以应对那些难以求解或解中有高频成分的挑战性问题，这导致了精度和收敛性的问题。这不仅会增加计算成本，还可能导致精度损失或解的发散。在本研究中，提出了一种替代方法来缓解上述问题。受到计算流体力学（CFD）社区中多重网格方法的启发，当前方法的基本思想是通过使用不同级别的训练样本进行训练，有效地去除不同频率的误差，从而以一种更简单的方式提高训练精度，而无需花费时间在神经网络结构、损失权重以及超参数的精细调整上。为了证明该方法的有效性，我们首先研究了一个具有高频成分的一维常微分方程（ODE）和一个使用V循环训练策略的二维对流扩散方程。最后，该方法被应用于经典的稳态基准问题——不同雷诺数下的驱动腔流动问题，以研究其在涉及多个高低频模式的问题中的适用性和有效性。通过利用各种训练序列模式，预测结果的改进使得30个案例可以探讨当前方法与迁移学习技术之间的协同作用，针对更复杂的问题（即更高的Re）。从目前的结果来看，该框架甚至能够在Re=5000的情况下产生良好的预测结果，展示了其解决复杂的高频PDE的能力。 显示英文摘要

摘要： Physics-Informed Neural Networks have emerged as a promising methodology for solving PDEs, gaining significant attention in computer science and various physics-related fields. Despite being demonstrated the ability to incorporate the physics of laws for versatile applications, PINNs still struggle with the challenging problems which are stiff to be solved and/or have high-frequency components in the solutions, resulting in accuracy and convergence issues. It may not only increase computational costs, but also lead to accuracy loss or solution divergence. In this study, an alternative approach is proposed to mitigate the above-mentioned problems. Inspired by the multi-grid method in CFD community, the underlying idea of the current approach is to efficiently remove different frequency errors via training with different levels of training samples, resulting in a simpler way to improve the training accuracy without spending time in fine-tuning of neural network structures, loss weights as well as hyperparameters. To demonstrate the efficacy of current approach, we first investigate canonical 1D ODE with high-frequency component and 2D convection-diffusion equation with V-cycle training strategy. Finally, the current method is employed for the classical benchmark problem of steady Lid-driven cavity flows at different Reynolds numbers, to investigate the applicability and efficacy for the problem involved multiple modes of high and low frequency. By virtue of various training sequence modes, improvement through predictions lead to 30% to 60% accuracy improvement. We also investigate the synergies between current method and transfer learning techniques for more challenging problems (i.e., higher Re). From the present results, it also revealed that the current framework can produce good predictions even for the case of Re=5000, demonstrating the ability to solve complex high-frequency PDEs.