用于三重周期极小曲面约束生成的傅里叶潜扩散模型
Fourier-Latent Diffusion for Constrained Generation of Triply Periodic Minimal Surfaces
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中文总结 AI 辅助
该研究提出基于扩散的生成框架，构建含超1.8万种TPMS的数据集，训练Transformer扩散模型实现TPMS的可控生成，满足几何与弹性能约束，为TPMS逆设计提供实用工具。
中文摘要 AI 辅助
我们提出一种基于扩散的生成框架，用于可控生成具有低残余均值的三重周期极小曲面（TPMS）结构。现有TPMS生成方法常局限于少数经典族，或生成偏离精确极小性的类TPMS近似结构。为构建该生成框架，我们首先通过在可镜像的基本边界体上枚举可允许边界环并求解多样的极小曲面片，构建了包含超过18000种独特TPMS的大规模数据集。随后，将每个曲面投影到紧凑的傅里叶潜空间，该空间明确施加周期性和$D_{2h}$对称性。接着，在该潜空间中训练基于Transformer的扩散模型，以支持无条件采样、确定性逆变换、局部编辑及用户指定约束下的条件生成。实验表明，该模型生成多样、低曲率的TPMS候选结构，在约束下满足稀疏几何约束并匹配目标均匀化线性弹性能，为TPMS逆设计提供实用工具。
英文摘要
We present a generative framework for the constrained design of $D_{2h}$-symmetric triply periodic minimal surfaces (TPMS) with low residual mean curvature. Existing TPMS design pipelines either explore a limited set of canonical analytical families or rely on computationally expensive numerical procedures, making it difficult to efficiently generate diverse candidates under geometric and/or mechanical requirements. Our approach learns a distribution of solver-generated TPMS in a compact Fourier space where periodicity and $D_{2h}$ symmetry are guaranteed by construction. This representation eliminates the need for a learned geometric decoder and enables the training of an effective diffusion model that generates low-mean-curvature candidates conditioned on sparse geometric constraints, selected homogenized elastic properties, or their combination. A subsequent coefficient-space refinement further reduces the residual mean curvature. Experiments show that our approach outperforms trigonometric, SDF-based, and standard Fourier-space baselines and enables controllable, high-quality generation under geometric and low-dimensional mechanical constraints. Overall, the proposed framework provides a compact and efficient design space for generating near-minimal periodic structures under user-specified requirements.
