We present a computational inverse design method for a new class of surface-based inflatable structure. Our deployable structures are fabricated by fusing together two layers of inextensible sheet material along carefully selected curves. The fusing curves form a network of tubular channels that can be inflated with air or other fluids. When fully inflated, the initially flat surface assumes a programmed double-curved shape and becomes stiff and load-bearing. We present a method that solves for the layout of air channels that, when inflated, best approximate a given input design. For this purpose, we integrate a forward simulation method for inflation with a gradient-based optimization algorithm that continuously adapts the geometry of the air channels to improve the design objectives. To initialize this non-linear optimization, we propose a novel surface flattening algorithm. When a channel is inflated, it approximately maintains its length, but contracts transversally to its main direction. Our algorithm approximates this deformation behavior by computing a mapping from the 3D design surface to the plane that allows for anisotropic metric scaling within the bounds realizable by the physical system. We show a wide variety of inflatable designs and fabricate several prototypes to validate our approach and highlight potential applications.