Improved randomized neural network methods with boundary processing for solving elliptic equations

We present two improved randomized neural network methods, namely the RNN-Scaling and RNN-Boundary-Processing (RNN-BP) methods, for solving elliptic equations such as the Poisson equation and the biharmonic equation. The RNNScaling method modifies the optimization objective by increasing the weight of boundary equations, resulting in a more accurate approximation. We propose the boundary processing techniques for the rectangular domain that enforce the RNN method to satisfy the non-homogeneous Dirichlet and clamped boundary conditions exactly. We further prove that the RNN-BP method is exact for solutions with specific forms and validate it numerically. Numerical experiments demonstrate that the RNN-BP method is the most accurate among the three methods, with the error reduced by up to 6 orders of magnitude for some tests.