Tensor Neural Network and Its Numerical Integration

In this paper, we introduce a type of tensor neural network. For the first time, we propose its numerical integration scheme and prove the computational complexity to be the polynomial scale of the dimension. Based on the tensor product structure, we design an efficient numerical integration method by using fixed quadrature points for the functions by the tensor neural network. The corresponding machine learning method is also introduced for solving high-dimensional eigenvalue problems. Some numerical examples are also provided to validate the theoretical results and the numerical algorithm.