  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.
