A piecewise extreme learning machine for interface problems

Deep learning methods have been developed to solve interface problems, benefiting from meshless features and the ability to approximate complex interfaces. However, existing deep neural network (DNN) methods for usual partial differential equations encounter accuracy limitations where after reaching a certain error level, further increases in network width, depth, and iteration steps do not enhance accuracy. This limitation becomes more notable in interface problems where the solution and its gradients may exhibit significant jumps across the interface. To improve accuracy, we propose a piecewise extreme learning machine (PELM) for addressing interface problems. An ELM is a type of shallow neural network where weight/bias coefficients in activation functions are randomly sampled and then fixed instead of being updated during the training process. Considering the solution jumps across the interface, we use a PELM scheme — setting one ELM function for each side of the interface. The two ELM functions are coupled using the interface conditions. Our numerical experiments demonstrate that the proposed PELM for the interface problem significantly improves the accuracy compared to conventional DNN solvers. The advantage of new method is shown for addressing interface problems that feature complex interface curves.