Adaptive deep neural networks for solving corner singular problems

Deep neural networks (DNNs) for numerical solutions to partial differential equations (PDEs) have exhibited their remarkable merits of meshless methods, dimensionless features, and nonlinear approximation powers. The majority of conventional DNNs are mainly concentrated on the low- or high-dimensional PDE with smooth solutions. In singular problems the DNN does not behave so efficiently as in the smooth problem because of low regularities of the solution. In this paper, we propose a novel adaptive DNN for the PDE with corner singular solutions. In the corner singular problem gradients of solution vary rapidly around some particular vertices or lines. This poses a difficulty in development of DNN solvers. We design adaptive DNN techniques for the corner singularity, which incorporates (a) an adaptive loss function that assigns radial weights to local losses, (b) an adaptive activation function that varies the choice of activation functions from neuron to neuron, and (c) an adaptive sampling strategy that selectively focuses on sampling points near the singular locations. The adaptive DNN with these adaptive schemes greatly enhances approximation accuracy and computation efficiency. This is verified by a comprehensive series of numerical experiments of 2D and 3D corner singular problems. The comparison analyzes with the existing DNN solvers are also made to demonstrate effectiveness of the new method. The L² relative errors of the proposed adaptive DNN are reduced by an order of magnitude with respect to several conventional DNN methods.