Deep Ritz Method for Elliptical Multiple Eigenvalue Problems

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 \(\mathcal {D}\) $\mathcal{D}$ , width \(\mathcal {W}\) $\mathcal{W}$ of the neural network, and training sample size n. By exploring the regularity of the EME and selecting an appropriate depth \(\mathcal {D}\) $\mathcal{D}$ and width \(\mathcal {W}\) $\mathcal{W}$ , we demonstrate that the desired bound enjoys a convergence rate of \(O(1/n^{16})\) $O(1/n^{16})$ , 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.