  In this paper, a new Discontinuity Capturing Shallow Neural Network (DCSNN)
for approximating $d$-dimensional piecewise continuous functions and for
solving elliptic interface problems is developed. There are three novel
features in the present network; namely, (i) jump discontinuities are
accurately captured, (ii) it is completely shallow, comprising only one hidden
layer, (iii) it is completely mesh-free for solving partial differential
equations. The crucial idea here is that a $d$-dimensional piecewise continuous
function can be extended to a continuous function defined in
$(d+1)$-dimensional space, where the augmented coordinate variable labels the
pieces of each sub-domain. We then construct a shallow neural network to
express this new function. Since only one hidden layer is employed, the number
of training parameters (weights and biases) scales linearly with the dimension
and the neurons used in the hidden layer. For solving elliptic interface
problems, the network is trained by minimizing the mean square error loss that
consists of the residual of the governing equation, boundary condition, and the
interface jump conditions. We perform a series of numerical tests to
demonstrate the accuracy of the present network. Our DCSNN model is efficient
due to only a moderate number of parameters needed to be trained (a few hundred
parameters used throughout all numerical examples), and the results indicate
good accuracy. Compared with the results obtained by the traditional grid-based
immersed interface method (IIM), which is designed particularly for elliptic
interface problems, our network model shows a better accuracy than IIM. We
conclude by solving a six-dimensional problem to demonstrate the capability of
the present network for high-dimensional applications.
