Randomized Radial Basis Function Neural Network for Solving Multiscale Elliptic 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 randomized radial basis function neural network 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 satisfying $C^0$ and $C^1$ 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 and the results demonstrate that the proposed method can improve the accuracy and efficiency well.