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

TL;DR

This paper introduces two enhanced randomized neural network methods, RNN-Scaling and RNN-Boundary-Processing, which improve the accuracy of solving elliptic equations by better boundary condition handling and optimization adjustments.

Contribution

The paper proposes novel boundary processing techniques and an improved optimization objective for randomized neural networks to solve elliptic equations more accurately.

Findings

RNN-BP achieves the highest accuracy among the methods.

Error reduction by 6 orders of magnitude in some tests.

Boundary conditions are satisfied exactly with RNN-BP.

Abstract

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