  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.
