After mapping, the mesh’s UV TextureCoordinates should be available on the mesh. Now, finding the 3d-coordinate for an arbitrary (u,v) value, I think you could do the following:

Duplicate the mesh in UV space, that is, assign the (u,v) coordinates to the physical coordinates like (u,v,0). Now you have a duplicate of the mesh in UV space. Then use a closest-point method to find the desired arbitrary UV-point’s face-index and the barycentric weights of that face.

Because the UV-mesh is a topological duplicate of the 3d mesh, you can get the 3d face using the same face index from the 3d mesh, then apply the barycentric weights to the face corners to compute the final 3d position.

I hope it makes sense