Funicular shells played a pivotal role in architecture and engineering both due to their aesthetic appeal and structural efficiency. The advent of modern techniques like 3D concrete printing has reignited interest in these forms, yet a fundamental challenge remains unsolved since Robert Hooke’s insight into the catenary arch in 1675: identifying the optimal three-dimensional shell shape that carries load purely through membrane compression, free from bending. Designing optimal shell structures is a complex process encompassing various factors, such as structural stability, load distribution, material properties, and geometric requirements. Significant progress has been made through computational methods and optimization algorithms, however, the high dimensionality and variety of applications make it difficult to find a universal solution. This work introduces a physics-informed generative framework for the optimal design of shell structures. The approach effectively frames design as an adversarial process, where a generator proposes shell geometries, while a discriminator, enhanced with physics-informed neural networks (PINNs), evaluates their structural efficiency via the membrane ratio. This metric penalizes bending-dominated forms, guiding the generation toward purely compressive, funicular solutions. The method inherently avoids complex optimization routines through the direct generation of diverse, physically valid shell structures even from non-optimal training data. Results demonstrate that the model can produce a range of efficient geometries, provide designers with tunable flexibility in the membrane behaviour, and allow smooth interpolation within the latent design space. The methodology is designed to address both inverse modeling scenarios and the generation of diverse, physically optimal shell geometries. By integrating AI with mechanical principles, this work offers a novel, generative tool that advances both the automation and creative potential in shell design for modern applications such as 3D concrete printing.