论文标题
CDNNS：耦合的深神经网络，用于Stokes和Darcy-Forchheimer问题的耦合
CDNNs: The coupled deep neural networks for coupling of the Stokes and Darcy-Forchheimer problems
论文作者
论文摘要
在本文中，我们提出了一种有效的深度学习方法，称为耦合深神经网络（CDNN），以解决耦合的物理问题。我们的方法将耦合PDE的界面条件正确地汇总到网络中，并可以作为复杂耦合问题的有效替代方法。为了施加节能约束，CDNN使用简单的完全连接的层和自定义损耗函数来执行模型训练过程以及精确解决方案的物理特性。由于以下原因，该方法可能是有益的：首先，我们随机采样，并且仅输入空间坐标而不会受样品性质的限制。其次，我们的方法是无误的，它比传统方法更有效。最后，我们的方法是平行的，可以同时独立解决多个变量。我们提供理论，以确保损失函数的收敛性以及神经网络与精确解决方案的收敛性。进行并讨论了一些数值实验，以证明该方法的性能。
In this article, we present an efficient deep learning method called coupled deep neural networks (CDNNs) for coupled physical problems. Our method compiles the interface conditions of the coupled PDEs into the networks properly and can be served as an efficient alternative to the complex coupled problems. To impose energy conservation constraints, the CDNNs utilize simple fully connected layers and a custom loss function to perform the model training process as well as the physical property of the exact solution. The approach can be beneficial for the following reasons: Firstly, we sampled randomly and only input spatial coordinates without being restricted by the nature of samples. Secondly, our method is meshfree which makes it more efficient than the traditional methods. Finally, our method is parallel and can solve multiple variables independently at the same time. We give the theory to guarantee the convergence of the loss function and the convergence of the neural networks to the exact solution. Some numerical experiments are performed and discussed to demonstrate the performance of the proposed method.
