  In this paper, we extend the class of kernel methods, the so-called diffusion
maps (DM) and ghost point diffusion maps (GPDM), to solve the time-dependent
advection-diffusion PDE on unknown smooth manifolds without and with
boundaries. The core idea is to directly approximate the spatial components of
the differential operator on the manifold with a local integral operator and
combine it with the standard implicit time difference scheme. When the manifold
has a boundary, a simplified version of the GPDM approach is used to overcome
the bias of the integral approximation near the boundary. The Monte-Carlo
discretization of the integral operator over the point cloud data gives rise to
a mesh-free formulation that is natural for randomly distributed points, even
when the manifold is embedded in high-dimensional ambient space. Here, we
establish the convergence of the proposed solver on appropriate topologies,
depending on the distribution of point cloud data and boundary type. We provide
numerical results to validate the convergence results on various examples that
involve simple geometry and an unknown manifold. Additionally, we also found
positive results in solving the one-dimensional viscous Burger's equation where
GPDM is adopted with a pseudo-spectral Galerkin framework to approximate
nonlinear advection term.
