Various intriguing morphological patterns emerge on the surfaces of growing, developing, or aging tissues, organs, and colonies of microorganisms. Our research draws inspiration from biological growth processes and applies differential growth to film-substrate systems. The strain mismatch induced by this approach leads to distinctive buckling patterns. Through a combination of finite element analysis (FEA) and theoretical analysis, we derive a comprehensive expression for the strain energy in a buckled film-substrate system subjected to differential growth. By solving the energy equations, our study investigates the intricate interplay of material properties, growth functions, and geometric characteristics. This exploration provides valuable insights into the deformations induced by growth in membrane structures and soft materials. Additionally, we establish a phase diagram that characterizes the buckling patterns. It opens avenues for innovative geometric design and 3D self-assembly techniques. Furthermore, our findings have the potential to inspire applications in fields such as soft robotics and flexible electronics.