We present a subspace method based on neural networks (SNN) for solving the partial differential equation with high accuracy. The basic idea of our method is to use some functions based on neural networks as base functions to span a subspace, then find an approximate solution in this subspace. We design two special algorithms in the strong form of partial differential equation. One algorithm enforces the equation and initial boundary conditions to hold on some collocation points, and another algorithm enforces $L^2$-norm of the residual of the equation and initial boundary conditions to be $0$. Our method can achieve high accuracy with low cost of training. Moreover, our method is free of parameters that need to be artificially adjusted. Numerical examples show that the cost of training these base functions of subspace is low, and only one hundred to two thousand epochs are needed for most tests. The error of our method can even fall below the level of $10^{-10}$ for some tests. The performance of our method significantly surpasses the performance of PINN and DGM in terms of the accuracy and computational cost.

翻译：我们提出一种基于神经网络的子空间方法（SNN），用于高精度求解偏微分方程。该方法的基本思想是利用基于神经网络的函数作为基函数张成子空间，然后在该子空间中寻找近似解。我们设计了两种基于偏微分方程强形式的特殊算法。一种算法强制方程及初始边界条件在某些配点上成立，另一种算法强制方程残差及初始边界条件的$L^2$范数为$0$。我们的方法能够以较低的训练成本实现高精度。此外，该方法无需人为调整任何参数。数值实验表明，训练这些子空间基函数的成本很低，大多数测试仅需100至2000个训练周期。在某些测试中，我们的方法误差甚至可降至$10^{-10}$量级。在精度和计算成本方面，本方法的性能显著优于PINN和DGM。