A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems. (arXiv:2106.05587v3 [math.NA] UPDATED)

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 …

arxiv discontinuity math network neural network