Designing and fabricating structures with specific mechanical properties requires understanding the intricate relationship between design parameters and performance. Understanding the design-performance relationship becomes increasingly complicated for nonlinear deformations. Though successful at modeling elastic deformations, simulation-based techniques struggle to model large elastoplastic deformations exhibiting plasticity and densification. We propose a neural network trained on experimental data to learn the design-performance relationship between 3D-printable shells and their compressive force-displacement behavior. Trained on thousands of physical experiments, our network aids in both forward and inverse design to generate shells exhibiting desired elastoplastic and hyperelastic deformations. We validate a subset of generated designs through fabrication and testing. Furthermore, we demonstrate the network's inverse design efficacy in generating custom shells for several applications.