Fig. 3: VAE-based latent space construction and optimization-driven inverse design.

a The attributed network embedding VAE model processes the truss lattice structure \({\mathsf{G}}=({\mathsf{A}},{\mathsf{x}})\), defined by the adjacency matrix \({\mathsf{A}}\) and feature vector \({\mathsf{x}}\). The adjacency encoder \({{\mathsf{E}}}_{{\mathsf{A}}\phi }\) and feature encoder \({{\mathsf{E}}}_{{\mathsf{x}}\phi }\) map \({\mathsf{G}}\) to a shared latent space \({\mathsf{z}}\), which follows a normal distribution. This reduced representation is passed to the adjacency decoder \({{\mathsf{D}}}_{{\mathsf{A}}\theta }\) and feature decoder \({{\mathsf{D}}}_{{\mathsf{x}}\theta }\) to reconstruct \({{\mathsf{G}}}^{{\prime} }=({{\mathsf{A}}}^{{\prime} },{{\mathsf{x}}}^{{\prime} })\). A property predictor \({{\mathsf{P}}}_{\omega }\) uses \({\mathsf{z}}\) to estimate the effective stiffness tensor \({\mathsf{C}}\) and nonlinear stress-strain response σ(ϵ) of the curved trusses. The figure shows the prediction accuracy for both properties. b Prediction performance is also evaluated specifically for \({\mathsf{C}}\). c A gradient-based optimization framework in the latent space \({\mathsf{z}}\) generates curved trusses with target properties. Starting with an initial guess, optimization searches for valid structures matching desired properties. The candidate structures are encoded, their latent representations predicted, and validity ensured. d Two examples demonstrate the process, starting from random latent points, optimizing to achieve target \({\mathsf{C}}\), and showing intermediate optimization steps.