Tensor Neural Network Interpolation and Its Applications

TL;DR

This paper introduces a tensor neural network-based interpolation method for high-dimensional non-tensor-product functions, enabling efficient numerical solutions for complex integrals and PDEs.

Contribution

It proposes a novel interpolation scheme that transforms non-tensor-product functions into tensor neural network representations, facilitating high-accuracy numerical computations.

Findings

Effective approximation of high-dimensional functions.

Improved numerical methods for high-dimensional integrals.

Validated through numerical examples.

Abstract

Based on tensor neural network, we propose an interpolation method for high dimensional non-tensor-product-type functions. This interpolation scheme is designed by using the tensor neural network based machine learning method. This means that we use a tensor neural network to approximate high dimensional functions which has no tensor product structure. In some sense, the non-tenor-product-type high dimensional function is transformed to the tensor neural network which has tensor product structure. It is well known that the tensor product structure can bring the possibility to design highly accurate and efficient numerical methods for dealing with high dimensional functions. In this paper, we will concentrate on computing the high dimensional integrations and solving high dimensional partial differential equations. The corresponding numerical methods and numerical examples will be provided to validate the proposed tensor neural network interpolation.