In this paper, based on the combination of tensor neural network and a posteriori error estimator, a novel type of machine learning method is proposed to solve high-dimensional boundary value problems with homogeneous and non-homogeneous Dirichlet or Neumann type of boundary conditions and eigenvalue problems of the second-order elliptic operator. The most important advantage of the tensor neural network is that the high dimensional integrations of tensor neural networks can be computed with high accuracy and high efficiency. Based on this advantage and the theory of a posteriori error estimation, the a posteriori error estimator is adopted to design the loss function to optimize the network parameters adaptively. The applications of tensor neural network and the a posteriori error estimator improve the accuracy of the corresponding machine learning method. The theoretical analysis and numerical examples are provided to validate the proposed methods.

翻译：本文基于张量神经网络与后验误差估计器的结合，提出了一种新型机器学习方法，用于求解齐次/非齐次Dirichlet或Neumann型边界条件的高维边值问题以及二阶椭圆算子的特征值问题。张量神经网络最重要的优势在于，其高维积分能够实现高精度与高效率的计算。基于这一优势并融合后验误差估计理论，本文采用后验误差估计器设计损失函数，从而自适应优化网络参数。张量神经网络与后验误差估计器的应用提升了相应机器学习方法的精度。理论分析与数值算例验证了所提方法的有效性。