A discontinuity capturing shallow neural network for elliptic interface problems

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