In this paper, a shallow Ritz-type neural network for solving elliptic equations with delta function singular sources on an interface is developed. There are three novel features in the present work; namely, (i) singularity naturally removed, (ii) level set introduced as feature input, (iii) it completely shallow, comprising only one hidden layer. We first introduce energy functional of problem...

The usual estimates for conjugate gradients (CG) specify a non–trivial rate of convergence right from the beginning. We investigate situations where the same can be said for Ritz values (considered as approximations to eigenvalues). We investigate the effect on the convergence behaviour of Ritz values of multiplying the weight functions by certain functions of polynomial growth. This will be sh...

By the semi-inverse method, a variational principle is obtained for the Thomas– Fermi equation, then the Ritz method is applied to solve an analytical solution, which is a much simpler and more efficient method. 2002 Elsevier Science Inc. All rights reserved.