AN ACCURATE AND EFFICIENT CONTINUITY-PRESERVED METHOD BASED ON RANDOMIZED NEURAL NETWORKS FOR ELLIPTIC INTERFACE PROBLEMS

In this paper, based on the extreme learning machine idea and randomized neural networks, a new continuity-preserved method is proposed to efficiently and accurately solve linear and nonlinear elliptic interface problems. For linear interface problems, an error estimate including an approximation error and a statistical error is established for shallow randomized neural networks with the activation function tanh(x). Especially, the approximation error under the L-infinity norm is given for the first time to the best of the authors' knowledge. For nonlinear cases, a novel multilevel method is further proposed to significantly improve efficiency. Various numerical tests on different examples are carried out to verify the proposed method, not only showing it can outperform classical numerical methods in terms of accuracy, but also illustrating its effectiveness, especially the improvements of the multilevel method in terms of computational costs.