Solving high-dimensional partial differential equations (PDEs) is a long-term computational challenge due to the fundamental obstacle known as the curse of dimensionality. This paper develops a novel method (DL4HPDE) based on residual neural network learning with data-driven learning elliptic PDEs on a box-shaped domain. However, to combine a strong mechanism with a weak mechanism, we reconstruct a trial solution to the equations in two parts: the first part satisfies the initial and boundary conditions, while the second part is the residual neural network algorithm, which is used to train the other part. In our proposed method, residual learning is adopted to make our model easier to optimize. Moreover, we propose a data-driven algorithm that can increase the training spatial points according to the regional error and improve the accuracy of the model. Finally, the numerical experiments show the efficiency of our proposed model.

由于称为维数诅咒的基本障碍，解决高维偏微分方程（PDE）是一项长期的计算挑战。本文开发了一种新方法（DL4HPDE）基于残差神经网络学习，并在盒形域上使用数据驱动的学习椭圆PDE。但是，为了将强机制与弱机制结合起来，我们分两部分重建了方程的试验解：第一部分满足初始条件和边界条件，第二部分是残差神经网络算法，用于训练另一部分。在我们提出的方法中，采用残差学习使我们的模型更易于优化。此外，我们提出了一种数据驱动算法，该算法可以根据区域误差增加训练空间点，并提高模型的准确性。最后，数值实验表明了我们提出的模型的有效性。