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 $\Gamma$-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.

翻译：我们提出了一种基于深层学习的一阶系统最低广场(FOSLS)方法(FOSLS),该方法基于对二阶椭圆形PDE进行数字解析的深层次学习。我们建议的方法能够处理变异和非变性问题,由于其无网状性质,它也可以处理高维域中的问题。我们证明了神经网络近似对于解决持续问题的一致程度,并将趋同证据推广到一些众所周知的相关方法。最后,我们提出了几个数字例子,说明我们离散性的表现。