A multilevel stochastic-gradient neural solver for boundary integral equations

Abstract

We develop a multilevel stochastic-gradient neural solver for boundary integral equations of the second kind. The unknown density is represented by a multilayer perceptron, trained by minimizing the Nyström-discretized residual on a ladder of refining quadrature grids, each level warm-started from the parameters of the previous one. Each step requires only dense matrix-vector products on mini-batches of collocation rows and network passes, operations that map directly onto GPU hardware. The residual contraction is governed by the empirical neural tangent kernel (NTK), the discrete sample of a single continuum kernel. On a fixed grid, training stalls once the residual concentrates in those modes at which the network contracts slowly. A spectral analysis, confirmed numerically, explains how refining the quadrature resolves more of the continuum kernel's spectrum and returns these modes to the optimizer's reach. Spectral bias, elsewhere an obstruction to neural network solvers, thus serves as the smoother of a multigrid-type iteration, with quadrature refinement in place of coarse-grid correction. The uniform conditioning of the discrete second-kind operator ensures that the NTK is the sole rate-determining spectrum, allowing the training residual to serve as an a posteriori error bound. Experiments on interior Dirichlet Laplace/Poisson problems and exterior Neumann Helmholtz problems {\color{black}{in ${\mathbb{R}}^d,d=2,3$}}, using both parametric and signed-distance surface representations, demonstrate the effectiveness and efficiency of the proposed method compared with GMRES at comparable tolerances. A final example, an exterior Robin problem for the Laplace equation on a ring-shaped hypersurface in ${\mathbb{R}}^4$, illustrates that the mesh-free formulation extends to higher-dimensional ambient spaces.