In this article, we propose three methods Power Method Neural Network (PMNN), Inverse Power Method Neural Networ (IPMNN) and Shifted Inverse Power Method Neural Network (SIPMNN) combined with power method, inverse power method and shifted inverse power method to solve eigenvalue problems with the dominant eigenvalue, the smallest eigenvalue and the smallest zero eigenvalue, respectively. The methods share similar spirits with traditional methods, but the differences are the differential operator realized by Automatic Differentiation (AD), the eigenfunction learned by the neural network and the iterations implemented by optimizing the specially defined loss function. We examine the applicability and accuracy of our methods in several numerical examples in high dimensions. Numerical results obtained by our methods for multidimensional problems show that our methods can provide accurate eigenvalue and eigenfunction approximations.
翻译:在本篇文章中,我们提出了三种方法:电动方法神经网络(PMNN)、反动力方法神经网络(IPMNN)和变换反动力方法神经网络(SIPMNN),结合动力方法、反动力方法,并转换反动力方法,分别解决占支配地位的六金价值、最小的六金价值和最小的零二元价值的二次价值问题。这些方法与传统方法有着相似的精神,但不同之处在于自动差异(AD)所实现的不同操作者、神经网络所学的二次元和通过优化特定损失功能所实施的迭代。我们用多个数字例子审视了我们的方法在高维度方面的适用性和准确性。我们用多维问题方法取得的数值结果表明,我们的方法可以提供准确的六金价值和无源组合。