Based on neural network and adaptive subspace approximation method, we propose a new machine learning method for solving partial differential equations. The neural network is adopted to build the basis of the finite dimensional subspace. Then the discrete solution is obtained by using the subspace approximation. Especially, based on the subspace approximation, a posteriori error estimator can be derivated by the hypercircle technique. This a posteriori error estimator can act as the loss function for adaptively refining the parameters of neural network.

翻译：基于神经网络与自适应子空间逼近方法，我们提出了一种求解偏微分方程的新型机器学习方法。该方法采用神经网络构建有限维子空间的基函数，进而通过子空间逼近获得离散解。特别地，基于子空间逼近，可利用超圆技术推导出后验误差估计量。该后验误差估计量可作为损失函数，用于自适应地优化神经网络的参数。