CH-WGAN: Uncertainty-Aware Generative Design of Three-Dimensional Aperiodic Architected Cellular Materials Via Signed Distance Fields

Abstract Metamaterial design (via architected cellular units) has evolved from patterning in a regular, periodic fashion to a random, aperiodic style to broaden the spectrum of the property space. While periodic designs excel at maximizing properties in specific directions (anisotropy), aperiodic or randomized designs are increasingly valued for their ability to homogenize effective properties (isotropy) and improve robustness against failure. However, this transition introduces geometric uncertainty—the stochastic variability inherent to the design and generation of aperiodic architected cellular materials (AACMs). The uncertainty in geometry cripples downstream workflows such as validation, optimization, and manufacturing. To manage this, state-of-the-art (SotA) strategies were developed to capture and incorporate randomness and enable statistical analysis in both traditional (e.g., probabilistic representations and uncertainty quantification) and data-driven (e.g., deep generative models and graph neural networks) manner. While traditional strategies often suffer from high computational cost and oversimplified assumptions about structure and uncertainty, data-driven approaches face challenges in interpretability and latent space entanglement. In contrast, we present an uncertainty-aware generative design framework that leverages a conditional hierarchical Wasserstein generative adversarial network (CH-WGAN) to synthesize diverse, high-fidelity 3D AACM by hypothesizing dependencies between periodic (nominal) cellular units and aperiodic (variant) ones. In CH-WGAN, a proposed parameter generator maps nominal control parameters and a dedicated aperiodicity code, together with latent noise, into a convex mixture of analytically defined base signed distance functions (SDFs), whose scale and periodicity distributions explicitly model geometric uncertainty. The critic, augmented with an InfoGAN-style Q-head, enforces a Wasserstein gradient penalty (GP) loss for realism, a latent-regression loss for invertibility, and entropy/Kullback–Leibler (KL) divergence to promote multimodal coverage of the uncertain design space. We introduce a periodicalization module that adaptively warps the underlying 3D grid according to learned periodicity distributions, enabling accurate aperiodicity modeling. Training on paired nominal-variant SDF data disentangles intrinsic geometry from uncertainty in periodicity, allowing CH-WGAN to generate multiple aperiodic, structurally diverse unit-cell variants for a periodic parent. Marching-cubes visualizations and statistical comparisons confirm that our learned uncertainty distribution closely matches real aperiodicity statistics. This framework provides a robust, interpretable, and generalizable tool for exploring AACM designs under geometric uncertainty.