标题： 一种混合形式的PINNS（MF-PINNS）用于求解耦合的Stokes-Darcy方程 显示英文标题

标题： A Mixed-Form PINNS (MF-PINNS) For Solving The Coupled Stokes-Darcy Equations

摘要： 并行物理信息神经网络（P-PINNs）已被广泛用于求解具有多个耦合物理场的系统，例如带有Beavers-Joseph-Saffman（BJS）界面条件的耦合Stokes-Darcy方程。 然而，偏微分方程（PDE）中过度高或低的物理常数通常会导致病态的损失函数，甚至可能导致PINNs训练数值解失败。 为了解决这个问题，本文开发了一种新型增强并行PINNs，即MF-PINNs。 我们的MF-PINNs将速度压力形式（VP）与流函数涡度形式（SV）相结合，并以调整后的权重添加到总损失函数中。 数值实验的结果表明，当运动粘度和渗透张量范围从1e-4到1e4时，我们的MF-PINNs成功提高了流线场和压力场的准确性。 因此，我们的MF-PINNs在涉及湍流的更复杂的PDE系统中具有前景。 此外，我们还探索了激活函数及其周期性的最佳组合。 我们还尝试设置初始学习率并设计其衰减策略。 与本文相关的代码和数据可在https://github.com/shxshx48716/MF-PINNs.git获取。 显示英文摘要

摘要： Parallel physical information neural networks (P-PINNs) have been widely used to solve systems with multiple coupled physical fields, such as the coupled Stokes-Darcy equations with Beavers-Joseph-Saffman (BJS) interface conditions. However, excessively high or low physical constants in partial differential equations (PDE) often lead to ill conditioned loss functions and can even cause the failure of training numerical solutions for PINNs. To solve this problem, we develop a new kind of enhanced parallel PINNs, MF-PINNs, in this article. Our MF-PINNs combines the velocity pressure form (VP) with the stream-vorticity form (SV) and add them with adjusted weights to the total loss functions. The results of numerical experiments show our MF-PINNs have successfully improved the accuracy of the streamline fields and the pressure fields when kinematic viscosity and permeability tensor range from 1e-4 to 1e4. Thus, our MF-PINNs hold promise for more chaotic PDE systems involving turbulent flows. Additionally, we also explore the best combination of the activation functions and their periodicity. And we also try to set the initial learning rate and design its decay strategies. The code and data associated with this paper are available at https://github.com/shxshx48716/MF-PINNs.git.