Deep Ritz Method for Elliptical Multiple Eigenvalue Problems

• Published in: Journal of scientific computing Vol. 98; no. 2; p. 48
• Main Authors: Ji, Xia, Jiao, Yuling, Lu, Xiliang, Song, Pengcheng, Wang, Fengru
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2024; Springer Nature B.V
• Subjects: Algorithms; Approximation; Artificial neural networks; Computational Mathematics and Numerical Analysis; Decomposition; Eigenvalues; Eigenvectors; Error analysis; Finite volume method; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Neural networks; Numerical analysis; Partial differential equations; Ritz method; Theoretical; Upper bounds; Convergence rate; 65N99; Elliptical multiple eigenvalue; DRM; FNN
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-023-02443-8

In this paper, we investigate solving the elliptical multiple eigenvalue (EME) problems using a Feedforward Neural Network. Firstly, we propose a general formulation for computing EME based on penalized variational forms of elliptical eigenvalue problems. Next, we solve the penalized variational form using the Deep Ritz Method. We establish an upper bound on the error between the estimated eigenvalues and true ones in terms of the depth D , width W of the neural network, and training sample size n . By exploring the regularity of the EME and selecting an appropriate depth D and width W , we demonstrate that the desired bound enjoys a convergence rate of O ( 1 / n 16 ) , which circumvents the curse of dimensionality. We also present several high-dimensional simulation results to illustrate the effectiveness of our proposed method and support our theoretical findings.