Shell structures played a pivotal role in the fields of architecture and engineering, due to their aesthetic appeal and structural efficiency. Once symbols of innovation, these structures became less prevalent in the last few decades due to a change in design movements and their complexity to build with conventional methods. In recent years, 3D concrete printing techniques have reignited the interest in these structures, nevertheless, a fundamental challenge remains unsolved since Robert Hooke’s insight into the catenary arch in 1675: the optimal three-dimensional shell shape that carries load purely through membrane compression. Optimal shell design is a complex process resulting from the interplay between a diversity of factors, such as: structural stability, material properties and geometric considerations. Significant progress has been made through computational methods and optimization algorithms. Nevertheless, the high dimensionality and variability of potential applications hinder the discovery of a universal solution. Generative Adversarial Networks (GANs), have shown remarkable success in generating realistic data samples matching the distribution of the training data. Their capabilities have been demonstrated across various domains, particularly in image-to-image translation, where they can produce highly convincing synthetic images. This work proposes a physics-informed generative framework for the optimal design of shell structures. The approach employs a GAN architecture guided by physical criteria to generate realistic and structurally efficient shell geometries. Specifically, the model is constrained by the membrane ratio, which quantifies the proportion of membrane energy relative to the total potential energy, to penalize geometries dominated by bending. The methodology is designed to address both inverse modeling scenarios and the generation of diverse, physically optimal shell geometries.