  The discovery of underlying surface partial differential equation (PDE) from
observational data has significant implications across various fields, bridging
the gap between theory and observation, enhancing our understanding of complex
systems, and providing valuable tools and insights for applications. In this
paper, we propose a novel approach, termed physical-informed sparse
optimization (PIS), for learning surface PDEs. Our approach incorporates both
$L_2$ physical-informed model loss and $L_1$ regularization penalty terms in
the loss function, enabling the identification of specific physical terms
within the surface PDEs. The unknown function and the differential operators on
surfaces are approximated by some extrinsic meshless methods. We provide
practical demonstrations of the algorithms including linear and nonlinear
systems. The numerical experiments on spheres and various other surfaces
demonstrate the effectiveness of the proposed approach in simultaneously
achieving precise solution prediction and identification of unknown PDEs.
