Randomized radial basis function neural network for solving multiscale elliptic equations

Abstract

Ordinary deep neural network (DNN)-based methods frequently encounter difficulties when tackling multiscale and high-frequency partial differential equations. To overcome these obstacles and improve computational accuracy and efficiency, this paper presents the Randomized Radial Basis Function Neural Network (RRNN), an innovative approach explicitly crafted for solving multiscale elliptic equations. The RRNN method commences by decomposing the computational domain into non-overlapping subdomains. Within each subdomain, the solution to the localized subproblem is approximated by a RRNN with a Gaussian kernel. This network is distinguished by the random assignment of width and center coefficients for its activation functions, thereby rendering the training process focused solely on determining the weight coefficients of the output layer. For each subproblem, similar to the Petrov–Galerkin finite element method, a linear system will be formulated on the foundation of a weak formulation. Subsequently, a selection of collocation points is stochastically sampled at the boundaries of the subdomain, ensuring the satisfaction of C⁰ and C¹ continuity and boundary conditions to couple these localized solutions. The network is ultimately trained using the least squares method to ascertain the output layer weights. To validate the RRNN method’s effectiveness, an extensive array of numerical experiments has been executed. The RRNN is firstly compared with a variety of DNN methods based on gradient descent optimization. The comparative analysis demonstrates the RRNN’s superior performance with respect to computational accuracy and training time. Furthermore, it is contrasted with to local extreme learning machine method, which also utilizes domain decomposition and the least squares method. The comparative findings suggest that the RRNN method can attain enhanced accuracy at a comparable computational cost, particularly pronounced in scenarios with a smaller scale ratio ɛ.