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