Inverse design of curved mechanical metamaterials with geometric AI: a generative diffusion operates in compact latent space of cellular structures

The inverse design of curved truss structures in mechanical metamaterials remains underexplored due to the vast, discrete design space, lack of structural representation, high dimensionality, and inversion ambiguity. This work introduces a geometric AI framework to address these challenges and enable the design of curved cellular structures with targeted effective properties. A graph-based representation is developed and used to generate a dataset of over 200,000 unique structures, combining stiff straight beams with compliant curved elements, guided by tetragonal symmetry. A joint-attributed network embedding variational autoencoder constructs a continuous latent space encoding both topology and geometry, enabling prediction of linear and nonlinear properties. The inverse problem is solved in latent space using gradient-based optimization and a diffusion model conditioned on linear properties. The diffusion model achieves higher accuracy and efficiency, offering a scalable, flexible approach for discovering structures with both compliant and ultra-stiff behaviors and tunable nonlinear responses.