  In this paper, we extend the Generalized Finite Difference Method (GFDM) on
unknown compact submanifolds of the Euclidean domain, identified by randomly
sampled data that (almost surely) lie on the interior of the manifolds.
Theoretically, we formalize GFDM by exploiting a representation of smooth
functions on the manifolds with Taylor's expansions of polynomials defined on
the tangent bundles. We illustrate the approach by approximating the
Laplace-Beltrami operator, where a stable approximation is achieved by a
combination of Generalized Moving Least-Squares algorithm and novel linear
programming that relaxes the diagonal-dominant constraint for the estimator to
allow for a feasible solution even when higher-order polynomials are employed.
We establish the theoretical convergence of GFDM in solving Poisson PDEs and
numerically demonstrate the accuracy on simple smooth manifolds of low and
moderate high co-dimensions as well as unknown 2D surfaces. For the Dirichlet
Poisson problem where no data points on the boundaries are available, we employ
GFDM with the volume-constraint approach that imposes the boundary conditions
on data points close to the boundary. When the location of the boundary is
unknown, we introduce a novel technique to detect points close to the boundary
without needing to estimate the distance of the sampled data points to the
boundary. We demonstrate the effectiveness of the volume-constraint employed by
imposing the boundary conditions on the data points detected by this new
technique compared to imposing the boundary conditions on all points within a
certain distance from the boundary, where the latter is sensitive to the choice
of truncation distance and require the knowledge of the boundary location.
