This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to approximate solutions of PDEs through the compositional construction and employs least-squares functionals as loss functions to determine parameters of the deep neural network. There are various least-squares functionals for a partial differential equation. This paper focuses on the so-called first-order system least-squares (FOSLS) functional studied in [3], which is based on a first-order system of scalar second-order elliptic PDEs. Numerical results for second-order elliptic PDEs in one dimension are presented.

本文研究了一种基于无监督深度学习的数值方法来求解偏微分方程（PDE）。该方法利用深层神经网络通过成分构造来近似PDE的解，并采用最小二乘函数作为损失函数来确定深层神经网络的参数。偏微分方程有多种最小二乘泛函。本文重点研究在文献[3]中研究的所谓一阶系统最小二乘（FOSLS）函数，该函数基于标量二阶椭圆PDE的一阶系统。给出了一维二阶椭圆形偏微分方程的数值结果。