# Flow Matching and Diffusion Models via PointNet for Generating Fluid Fields on Irregular Geometries

arXiv: https://arxiv.org/abs/2601.03030 · HTML: https://mlsanity.com/paper/2601.03030/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 47 of its 74 references that are in the map.
- First posted on arXiv 6 Jan 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, 30 of 47)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [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)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [On the Opportunities and Risks of Foundation Models](https://mlsanity.com/paper/2108.07258/) (2021)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Point Transformer](https://mlsanity.com/paper/2011.00931/) (2020)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [Diffusion Probabilistic Models for 3D Point Cloud Generation](https://mlsanity.com/paper/2103.01458/) (2021)
- [Point-E: A System for Generating 3D Point Clouds from Complex Prompts](https://mlsanity.com/paper/2212.08751/) (2022)
- [LION: Latent Point Diffusion Models for 3D Shape Generation](https://mlsanity.com/paper/2210.06978/) (2022)
- [Rethinking Network Design and Local Geometry in Point Cloud: A Simple Residual MLP Framework](https://mlsanity.com/paper/2202.07123/) (2022)
- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)

## Built on by (most cited first, 2 of 2)

- [One-Step Flow Matching for Generative Modeling of Path-Dependent Physical Fields](https://mlsanity.com/paper/2606.22752/) (2026)
- [Conditional Flow Matching for ML-Based Inverse Design Problems](https://mlsanity.com/paper/2609.00863/) (2026)

## Abstract (arXiv)

We present two novel generative geometric deep learning frameworks, termed Flow Matching PointNet and Diffusion PointNet, for predicting fluid flow variables on irregular geometries by incorporating PointNet into flow matching and diffusion models, respectively. In these frameworks, a reverse generative process reconstructs physical fields from standard Gaussian noise conditioned on unseen geometries. The proposed approaches operate directly on point-cloud representations of computational domains (e.g., grid vertices of finite-volume meshes) and therefore avoid the limitations of pixelation used to project geometries onto uniform lattices, as is common in U-Net-based flow matching and diffusion models. In contrast to graph neural network-based diffusion models, Flow Matching PointNet and Diffusion PointNet do not exhibit high-frequency noise artifacts in the predicted fields. Moreover, unlike such approaches, which require auxiliary intermediate networks to condition geometry, the proposed frameworks rely solely on PointNet, resulting in a simple and unified architecture. The performance of the proposed frameworks is evaluated on steady incompressible flow past a cylinder, using a geometric dataset constructed by varying the cylinder's cross-sectional shape and orientation across samples. The results demonstrate that Flow Matching PointNet and Diffusion PointNet achieve more accurate predictions of velocity and pressure fields, as well as lift and drag forces, and exhibit greater robustness to incomplete geometries compared to a vanilla PointNet with the same number of trainable parameters.