Before revision:  We  validate the proposed mesh-free PDE solver with supporting numerical examples on various problems , defined on simple manifolds embedded in 2D to 5D Euclidean spaces as well as on an unknown manifold.
After revision:  We validate the proposed mesh-free PDE solver with supporting numerical examples on various problems defined on simple manifolds embedded in 2D to 5D Euclidean spaces as well as on an unknown manifold.