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.

翻译：基于张量神经网络，我们提出了一种针对高维非张量积型函数的插值方法。该插值方案通过基于张量神经网络的机器学习方法设计，即利用张量神经网络来逼近不具有张量积结构的高维函数。从某种意义上说，非张量积型高维函数被转化为具有张量积结构的张量神经网络。众所周知，张量积结构能够为处理高维函数提供高精度、高效率数值方法设计的可能性。本文将重点研究高维积分计算和高维偏微分方程求解问题，并提供相应的数值方法和数值算例以验证所提出的张量神经网络插值方法。