Tensor Neural Network and Its Numerical Integration

TL;DR

This paper introduces a tensor neural network with a novel numerical integration scheme that is computationally efficient for high-dimensional problems, supported by theoretical analysis and numerical validation.

Contribution

It proposes the first numerical integration method for tensor neural networks, demonstrating polynomial complexity and applicability to high-dimensional machine learning tasks.

Findings

Numerical integration scheme has polynomial complexity.

Method effectively solves high-dimensional problems.

Numerical examples validate theoretical and computational results.

Abstract

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.