论文标题
深层的一阶系统最小二乘法，用于求解椭圆PDES
A deep first-order system least squares method for solving elliptic PDEs
论文作者
论文摘要
我们提出了一种基于数值求解二阶椭圆PDE的深度学习的一阶系统最小二乘方法（FOSLS）方法。我们提出的方法能够处理变分和非不同问题，并且由于其无网状性质，它也可以处理在高维域中构成的问题。我们证明了神经网络近似对连续问题解决方案的$γ$ - 连接，并将收敛证明扩展到一些众所周知的相关方法。最后，我们提出了几个数字示例，说明了我们离散化的性能。
We propose a First-Order System Least Squares (FOSLS) method based on deep-learning for numerically solving second-order elliptic PDEs. The method we propose is capable of dealing with either variational and non-variational problems, and because of its meshless nature, it can also deal with problems posed in high-dimensional domains. We prove the $Γ$-convergence of the neural network approximation towards the solution of the continuous problem, and extend the convergence proof to some well-known related methods. Finally, we present several numerical examples illustrating the performance of our discretization.
