A mesh-free method using piecewise deep neural network for elliptic interface problems

In this paper, we propose a novel mesh-free numerical method for solving the elliptic interface problems based on deep learning. We approximate solution by neural networks and, since may change dramatically across interface, employ different each sub-domain. By reformulating problem as least-squares problem, discretize objective function using mean squared error via sampling and solve proposed standard training algorithms such stochastic gradient descent. The discretized utilizes only point-wise information points thus no underlying mesh is required. Doing circumvents challenging meshing procedure well integration complex interfaces. To improve computational efficiency more problems, further design an adaptive strategy residual of algorithm. Finally, present several experiments in both 2D 3D to show flexibility, effectiveness, accuracy least-square problems.