Boundary neuron method for solving partial differential equations

TL;DR

The paper introduces a boundary neuron method with random features (BNM-RF) for efficiently solving PDEs by approximating boundary functions with shallow networks, avoiding complex optimization.

Contribution

It presents a novel neural network approach that combines boundary integral equations with random features, enabling efficient PDE solutions on complex geometries.

Findings

Achieves competitive accuracy with boundary element methods.

Performs well on Laplace and Helmholtz problems.

Requires only a few neurons for good performance.

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.