Boundary neuron method for solving partial differential equations

Abstract

We propose a boundary neuron method with random features (BNM-RF) for solving partial differential equations. The method approximates the unknown boundary function by a shallow network within the boundary integral formulation. With randomly sampled and fixed hidden parameters, the computation reduces to a linear least squares problem for the output coefficients, which avoids gradient based nonconvex optimization. This construction retains the dimensionality reduction of boundary integral equations and the linear solution structure of the random feature method. For elliptic problems, we establish convergence analysis by combining kernel-based method with random feature approximation, and obtain error bounds on both the boundary and the interior solution. Numerical experiments on Laplace and Helmholtz problems, including interior and exterior cases, show that the proposed method achieves competitive accuracy relative to the boundary element method and favorable performance relative to boundary integral neural networks in the tested settings with only few neurons. Overall, the proposed method provides a practical framework for combining boundary integral equations with neural network for problems on complex geometries and unbounded domains.

Figures (5)

• Figure 1: Numerical results for the case $k$=9 with $N_s$=60 and $M$=40: (a) true solution; (b) predicted solution; (c) error distribution.
• Figure 2: Illustrates the training dynamics through three subplots: (a) variation of the error with the number of collocation points with 40 fixed neurons for different k values; (b) variation of the error with the number of neurons under 60 fixed collocation points for different k values; (c) variation of the error with the number of collocation points for different k values with bem.
• Figure 3: Comparison of the predicted solutions and error plots of Laplace equation \ref{['eq:laplace_eq_1']} with Dirichlet boundary condition by BEM, BNM-RF, and BINN methods.
• Figure 4: Numerical results for Helmholtz equation \ref{['eq:origin_helmholtz']} with $k=2$ and Dirichlet boundary condition.
• Figure 5: Three-dimensional representations of numerical solutions on cross-sectional planes (Y-Z, X-Z, and X-Y) for plane wave scattering by a rigid sphere with wavenumber $k=2$ and $M=32$ neurons, obtained by BEM and BNM-RF.

Theorems & Definitions (7)

• Theorem 4.1
• proof
• Theorem 4.2: Theorem 2,liao2025solving
• Theorem 4.3
• proof
• Proposition 1: Sobolev sampling inequality on manifolds fuselier2012scattered
• Theorem B.1: Theorem 3.1,batlle2025error