An immersed interface neural network for elliptic interface problems

In this paper, a new immersed interface neural network (IINN) is proposed for solving interface problems in a regular domain with jump discontinuities on an embedded irregular interface. This method is introduced in Poisson interface problems, which can also be generalized to solving Stokes interface problems and elliptic interface problems. The main idea is using neural network to approximate the extension of the known jump conditions along the normal lines of the interface and constructing a discontinuity capturing function. With such function, the interface problem with a non-smooth solution can be changed to the problem with a smooth solution. The numerical result is composed of the discontinuity capturing function and the smooth solution. There are four novel features in the present work: (i) the jump discontinuities can be accurately captured; (ii) it is not required to label the mesh around the interface and finding the correction term like Immersed Interface Method (IIM); (iii) it is completely mesh-free for training the discontinuity capturing function; (iv) it preserves second-order accuracy for the solution. The numerical results show that the IINN is comparable and behaves better than the traditional immersed interface method and other neural network methods.