Extreme learning machine with a domain shifting separation strategy for interface problems

The physics-informed neural network (PINN) has attracted considerable interest in solving partial differential equations, including interface problems. Since solutions of the interface problem are generally (weakly) discontinuous along the interface, a piecewise PINN scheme has been adopted to improve approximation accuracy. There is a computational complexity to operate several piecewise neural networks in the PINN structure. This study investigates the feasibility to use a single neural network to solve the interface problem with multiple interfaces. To this end, we employ a domain shifting separation strategy. This strategy shifts sub-domains cut by the interfaces into separate and distinct domains, and the interfaces serve as boundaries of these domains. In this way, each original point on the interface is associated with two spatially distinct points on the boundaries of two domains, and the data exchange at the original point is maintained by imposing the exchange conditions at these two points. As a consequence, a single network can be used to combine the distinct domains. A shallow neural network, extreme learning machine, based on such a domain shifting scheme is developed to improve the accuracy. We apply the new method to both linear and nonlinear elliptic problems, as well as time-dependent linear and nonlinear interface problems, including those involving moving interfaces. For the time-dependent interface problem, the time dimension is treated as an additional spatial coordinate, and the domain shifting procedure becomes the time–space domain shifting separation. Numerical experiments illustrate that the proposed method markedly enhances the accuracy compared with traditional deep neural network solvers.