# boundary integral equations

Lu and Tsai propose a multilevel stochastic-gradient neural solver (MLSG) for second-kind boundary integral equations that represents the unknown boundary density with a neural network optimized by stochastic residual minimization over a hierarchy of successively refined Nyström discretizations, warm-starting each level with the previous level's network parameters; the algorithm avoids grid-transfer operators and hierarchical fast-summation machinery, relying instead on batched kernel evaluations and standard network forward and backward passes, and is demonstrated on Laplace/Poisson and Helmholtz problems in two and three dimensions plus an exterior Robin problem on a hypersurface in R⁴, under both parametric and signed-distance surface representations at up to million-scale discretizati

Lu and Tsai propose a multilevel stochastic-gradient neural solver (MLSG) for second-kind boundary integral equations that represents the unknown boundary density with a neural network optimized by stochastic residual minimization over a hierarchy of successively refined Nyström discretizations, warm-starting each level with the previous level's network parameters; the algorithm avoids grid-transfer operators and hierarchical fast-summation machinery, relying instead on batched kernel evaluations and standard network forward and backward passes, and is demonstrated on Laplace/Poisson and Helmholtz problems in two and three dimensions plus an exterior Robin problem on a hypersurface in R⁴, under both parametric and signed-distance surface representations at up to million-scale discretizati