Solving PDEs on Unknown Manifolds with Machine Learning

arXiv:2106.06682v4 Announce Type: replace-cross
Abstract: This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated as a supervised learning task to solve a least-squares regression problem that imposes an algebraic equation approximating a PDE (and boundary conditions if applicable). This algebraic equation involves a graph-Laplacian type matrix obtained via DM asymptotic expansion, which is a …

abstract arxiv computational cs.lg cs.na deep learning diffusion framework free least machine machine learning maps math.na mesh paper regression solve solver squares supervised learning theory type