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

arXiv: https://arxiv.org/abs/2608.02151 · HTML: https://mlsanity.com/paper/2608.02151/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 28 of its 60 references that are in the map.
- First posted on arXiv 3 Aug 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 28 of 28)

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- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Structured 3D Latents for Scalable and Versatile 3D Generation](https://mlsanity.com/paper/2412.01506/) (2024)
- [Learning a Probabilistic Latent Space of Object Shapes via 3D Generative-Adversarial Modeling](https://mlsanity.com/paper/1610.07584/) (2016)
- [All are Worth Words: A ViT Backbone for Diffusion Models](https://mlsanity.com/paper/2209.12152/) (2022)
- [PointFlow: 3D Point Cloud Generation with Continuous Normalizing Flows](https://mlsanity.com/paper/1906.12320/) (2019)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [SDFusion: Multimodal 3D Shape Completion, Reconstruction, and Generation](https://mlsanity.com/paper/2212.04493/) (2022)
- [MeshDiffusion: Score-based Generative 3D Mesh Modeling](https://mlsanity.com/paper/2303.08133/) (2023)
- [Diffusion-SDF: Conditional Generative Modeling of Signed Distance Functions](https://mlsanity.com/paper/2211.13757/) (2022)
- [Diffusion-SDF: Text-to-Shape via Voxelized Diffusion](https://mlsanity.com/paper/2212.03293/) (2022)
- [SDF-StyleGAN: Implicit SDF-Based StyleGAN for 3D Shape Generation](https://mlsanity.com/paper/2206.12055/) (2022)
- [Adversarial Generation of Continuous Implicit Shape Representations](https://mlsanity.com/paper/2002.00349/) (2020)
- [DCTdiff: Intriguing Properties of Image Generative Modeling in the DCT Space](https://mlsanity.com/paper/2412.15032/) (2024)
- [Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraints](https://mlsanity.com/paper/2506.08604/) (2025)
- [Neural Wavelet-domain Diffusion for 3D Shape Generation, Inversion, and Manipulation](https://mlsanity.com/paper/2302.00190/) (2023)
- [Geometry-Informed Neural Networks](https://mlsanity.com/paper/2402.14009/) (2024)
- [MIND: Microstructure INverse Design with Generative Hybrid Neural Representation](https://mlsanity.com/paper/2502.02607/) (2025)
- [An Optimized, Easy-to-use, Open-source GPU Solver for Large-scale Inverse Homogenization Problems](https://mlsanity.com/paper/2301.08911/) (2023)
- [A Mesh Is Worth 512 Numbers: Spectral-domain Diffusion Modeling for High-dimension Shape Generation](https://mlsanity.com/paper/2503.06485/) (2025)

## Abstract (arXiv)

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.