  In this paper, we introduce a tensor neural network based machine learning
method for solving the elliptic partial differential equations with random
coefficients in a bounded physical domain. With the help of tensor product
structure, we can transform the high-dimensional integrations of tensor neural
network functions to one-dimensional integrations which can be computed with
the classical quadrature schemes with high accuracy. The complexity of its
calculation can be reduced from the exponential scale to a polynomial scale.
The corresponding machine learning method is designed for solving
high-dimensional parametric elliptic equations. Some numerical examples are
provided to validate the accuracy and efficiency of the proposed algorithms.
