AI and the World's Hard Problems · Mathematics · arXiv · 2026-10-20
A multilevel stochastic-gradient neural solver for boundary integral equations
The paper proposes MLSG, a multilevel stochastic-gradient neural solver for second-kind boundary integral equations. A neural network represents the unknown boundary density and is optimized by stochastic residual minimization over a hierarchy of successively refined Nyström discretizations, with each new level warm-started from the previous one. This coarse-to-fine strategy keeps a continuous, grid-independent density representation while aiming to reduce computational effort, and experiments on Laplace/Poisson and Helmholtz problems in two and three dimensions, plus an exterior Robin problem on a hypersurface in R⁴, demonstrate the method at up to million-scale discretizations.