In this paper, we propose a type of tensor-neural-network-based machine learning method to compute multi-eigenpairs of high dimensional eigenvalue problems without Monte-Carlo procedure. Solving multi-eigenvalues and their corresponding eigenfunctions is one of the basic tasks in mathematical and computational physics. With the help of tensor neural network and deep Ritz method, the high dimensional integrations included in the loss functions of the machine learning process can be computed with high accuracy. The high accuracy of high dimensional integrations can improve the accuracy of the machine learning method for computing multi-eigenpairs of high dimensional eigenvalue problems. Here, we introduce the tensor neural network and design the machine learning method for computing multi-eigenpairs of the high dimensional eigenvalue problems. The proposed numerical method is validated with plenty of numerical examples.

翻译：本文提出一种基于张量神经网络的机器学习方法，用于计算高维特征值问题中的多个特征对，且无需蒙特卡洛过程。求解多个特征值及其对应的特征函数是数学与计算物理学中的基本任务之一。借助张量神经网络和深度里茨方法，机器学习过程中损失函数所包含的高维积分能够以高精度计算。高维积分的高精度可提升机器学习方法求解高维特征值问题中多个特征对的准确性。本文引入张量神经网络，并设计了用于计算高维特征值问题中多个特征对的机器学习方法。通过大量数值算例验证了所提出数值方法的有效性。