BEPGN: A boundary element-enhanced physics-informed graph neural network for solving Laplace boundary value problems

This paper proposes a Boundary Element-Enhanced Physics-Informed Graph Neural Network method (BEPGN), which integrates the Boundary Element Method, Physics-Informed Neural Networks, and Graph Neural Networks. Unlike existing BEM-based neural network solvers, which rely on feedforward networks and treat boundary elements independently, BEPGN constructs a graph from the discretized boundary and employs message passing to explicitly capture the topological relationships between adjacent elements, with the boundary integral equation residual serving directly as the physics-informed loss. Numerical experiments under Dirichlet, Neumann, and mixed boundary conditions demonstrate that BEPGN achieves accuracy comparable to traditional BEM and converges faster than the Boundary Element Method based Physics-Informed Neural Network approach (BEM-PINN), validating the advantage of graph-structured learning in boundary element-based physics-informed solvers.