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

TL;DR

This paper introduces a neural network-based subspace method for solving PDEs in weak form, achieving high accuracy with low training cost and demonstrating superior performance in numerical tests.

Contribution

It proposes a novel neural network subspace approach for PDEs that separates base function training from solution approximation, enhancing efficiency and accuracy.

Findings

Error below 10^{-7} in some tests

Requires only 100-2000 training epochs

Outperforms existing methods in accuracy and cost

Abstract

We present a subspace method based on neural networks for solving the partial differential equation in weak form 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. Training base functions and finding an approximate solution can be separated, that is different methods can be used to train these base functions, and different methods can also be used to find an approximate solution. In this paper, we find an approximate solution of the partial differential equation in the weak form. Our method can achieve high accuracy with low cost of training. Numerical examples show that the cost of training these base functions is low, and only one hundred to two thousand epochs are needed for most tests. The error of our method can fall below the level of $10^{-7}$ for some tests. The proposed method has the better performance in terms of the accuracy and computational cost.