Fourier-Latent Diffusion for Constrained Generation of Triply Periodic Minimal Surfaces

We propose a diffusion-based generative framework for controllable generation of triply periodic minimal surface (TPMS) structures with low residual mean curvature. Existing TPMS generation approaches are often restricted to a small set of canonical families or produce TPMS-like approximations that deviate from exact minimality. To enable this generative framework, we first construct a large-scale dataset of over 18K unique TPMS by enumerating admissible boundary loops on mirrorable fundamental bounding volumes and solving for diverse minimal-surface patches. Each surface is then projected onto a compact Fourier latent space that explicitly enforces periodicity and D2h symmetry. Next, a transformer-based diffusion model is trained in this latent space to support unconditional sampling, deterministic inversion, local editing, and conditional generation under user-specified constraints. Experiments demonstrate that the model generates diverse, low-curvature TPMS candidates that, under conditioning, satisfy sparse geometric constraints and match target homogenized linear elastic properties, providing a practical tool for TPMS inverse design.