  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 consistent
estimator of second-order elliptic differential operators. The resulting
numerical method is to solve a highly non-convex empirical risk minimization
problem subjected to a solution from a hypothesis space of neural networks
(NNs). In a well-posed elliptic PDE setting, when the hypothesis space consists
of neural networks with either infinite width or depth, we show that the global
minimizer of the empirical loss function is a consistent solution in the limit
of large training data. When the hypothesis space is a two-layer neural
network, we show that for a sufficiently large width, gradient descent can
identify a global minimizer of the empirical loss function. Supporting
numerical examples demonstrate the convergence of the solutions, ranging from
simple manifolds with low and high co-dimensions, to rough surfaces with and
without boundaries. We also show that the proposed NN solver can robustly
generalize the PDE solution on new data points with generalization errors that
are almost identical to the training errors, superseding a Nystrom-based
interpolation method.
