Subspace method based on neural networks for solving the partial differential equation

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.