  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) the delta
function singularity is naturally removed, (ii) level set function is
introduced as a feature input, (iii) it is completely shallow, comprising only
one hidden layer. We first introduce the energy functional of the problem and
then transform the contribution of singular sources to a regular surface
integral along the interface. In such a way, the delta function singularity can
be naturally removed without introducing a discrete one that is commonly used
in traditional regularization methods, such as the well-known immersed boundary
method. The original problem is then reformulated as a minimization problem. We
propose a shallow Ritz-type neural network with one hidden layer to approximate
the global minimizer of the energy functional. As a result, the network is
trained by minimizing the loss function that is a discrete version of the
energy. In addition, we include the level set function of the interface as a
feature input of the network and find that it significantly improves the
training efficiency and accuracy. We perform a series of numerical tests to
show the accuracy of the present method and its capability for problems in
irregular domains and higher dimensions.
