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

Huifang Zhou, Zhiqiang Sheng

TL;DR

This work tackles solving elliptic PDEs, notably the Poisson and biharmonic equations, with randomized neural networks whose weights are fixed. It introduces two boundary-focused enhancements: RNN-Scaling, which amplifies the importance of boundary equations in the LS system, and RNN-BP, which enforces Dirichlet and clamped boundary conditions exactly via boundary construction and interpolation. The authors prove exactness of RNN-BP for certain solution forms and demonstrate through extensive numerical experiments that RNN-BP and RNN-Scaling substantially outperform the baseline RNN, with error reductions reaching up to 6–9 orders of magnitude in some cases. The methods show robust performance across rectangular and circular domains, offering a practical, boundary-aware alternative to traditional mesh-based and PINN approaches in scientific computing.

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.

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

TL;DR

This work tackles solving elliptic PDEs, notably the Poisson and biharmonic equations, with randomized neural networks whose weights are fixed. It introduces two boundary-focused enhancements: RNN-Scaling, which amplifies the importance of boundary equations in the LS system, and RNN-BP, which enforces Dirichlet and clamped boundary conditions exactly via boundary construction and interpolation. The authors prove exactness of RNN-BP for certain solution forms and demonstrate through extensive numerical experiments that RNN-BP and RNN-Scaling substantially outperform the baseline RNN, with error reductions reaching up to 6–9 orders of magnitude in some cases. The methods show robust performance across rectangular and circular domains, offering a practical, boundary-aware alternative to traditional mesh-based and PINN approaches in scientific computing.

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.
Paper Structure (23 sections, 4 theorems, 50 equations, 19 figures, 8 tables)

This paper contains 23 sections, 4 theorems, 50 equations, 19 figures, 8 tables.

Table of Contents

  1. Introduction
  2. The improved RNN methods for the Poisson equation
  3. Randomized neural networks
  4. RNN method
  5. RNN-Scaling method
  6. RNN-BP method
  7. The improved RNN methods for the biharmonic equation
  8. RNN method and RNN-Scaling method
  9. RNN-BP method
  10. Numerical experiments
  11. Poisson equation
  12. Example 1.1
  13. Example 1.2
  14. Example 1.3
  15. Example 1.4
  16. ...and 8 more sections

Key Result

Theorem 1

The numerical solution u_RNN_BP_Poisson of the RNN-BP method satisfies the Dirichlet boundary condition exactly.

Figures (19)

  • Figure 1: The architecture of the RNN.
  • Figure 2: The absolute error of the RNN method.
  • Figure 3: The absolute error of the RNN-Scaling method.
  • Figure 4: The architecture of the RNN-BP.
  • Figure 5: Comparison of relative $L^2$ errors of three methods for Example 1.1: (a) Varying numbers of points with the default initialization. (b) Varying numbers of points with uniform random initialization ($R_m = 1$). (c) Varying max magnitude of random coefficients.
  • ...and 14 more figures

Theorems & Definitions (11)

  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 3
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 1 more