  Smooth and curved microstructural topologies found in nature - from soap
films to trabecular bone - have inspired several mimetic design spaces for
architected metamaterials and bio-scaffolds. However, the design approaches so
far have been ad hoc, raising the challenge: how to systematically and
efficiently inverse design such artificial microstructures with targeted
topological features? Here, we explore surface curvature as a design modality
and present a deep learning framework to produce topologies with as-desired
curvature profiles. The inverse design framework can generalize to diverse
topological features such as tubular, membranous, and particulate features.
Moreover, we demonstrate successful generalization beyond both the design and
data space by inverse designing topologies that mimic the curvature profile of
trabecular bone, spinodoid topologies, and periodic nodal surfaces for
application in bio-scaffolds and implants. Lastly, we bridge curvature and
mechanics by showing how topological curvature can be designed to promote
mechanically beneficial stretching-dominated deformation over bending-dominated
deformation.
