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 develop an efficient numerical integration method by using fixed quadrature points for the functions of the tensor neural network. The corresponding machine learning method is also introduced for solving high-dimensional problems. Some numerical examples are also provided to validate the theoretical results and the numerical algorithm.

翻译：本文介绍了一种张量神经网络。我们首次提出其数值积分方案，并证明计算复杂度为维度的多项式量级。基于张量积结构，我们利用固定求积点针对张量神经网络函数开发了一种高效的数值积分方法。同时引入相应的机器学习方法以求解高维问题。文中还提供了若干数值算例，以验证理论结果与数值算法。