# SE(3) symmetry lets graph neural networks learn arterial velocity estimation from small datasets

arXiv: https://arxiv.org/abs/2302.08780 · HTML: https://mlsanity.com/paper/2302.08780/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 30 references that are in the map.
- First posted on arXiv 17 Feb 2023.

## Builds on (most cited first, 8 of 8)

- [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)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Scalable algorithms for physics-informed neural and graph networks](https://mlsanity.com/paper/2205.08332/) (2022)
- [Mesh convolutional neural networks for wall shear stress estimation in 3D artery models](https://mlsanity.com/paper/2109.04797/) (2021)

## Built on by (most cited first, 11 of 11)

- [Learning Reduced-Order Models for Cardiovascular Simulations with Graph Neural Networks](https://mlsanity.com/paper/2303.07310/) (2023)
- [LaB-GATr: geometric algebra transformers for large biomedical surface and volume meshes](https://mlsanity.com/paper/2403.07536/) (2024)
- [Physics-informed graph neural networks for flow field estimation in carotid arteries](https://mlsanity.com/paper/2408.07110/) (2024)
- [Deep vectorised operators for pulsatile hemodynamics estimation in coronary arteries from a steady-state prior](https://mlsanity.com/paper/2410.11920/) (2024)
- [Domain independent post-processing with graph U-nets: Applications to Electrical Impedance Tomographic Imaging](https://mlsanity.com/paper/2305.05020/) (2023)
- [Learning Hemodynamic Scalar Fields on Coronary Artery Meshes: A Benchmark of Geometric Deep Learning Models](https://mlsanity.com/paper/2501.09046/) (2025)
- [Active Learning for Deep Learning-Based Hemodynamic Parameter Estimation](https://mlsanity.com/paper/2503.03453/) (2025)
- [Wall Shear Stress Estimation in Abdominal Aortic Aneurysms: Towards Generalisable Neural Surrogate Models](https://mlsanity.com/paper/2507.22817/) (2025)
- [Mesh-Informed Reduced Order Models for Aneurysm Rupture Risk Prediction](https://mlsanity.com/paper/2410.03802/) (2024)
- [When Does Equivariance Help? Canonical Alignment in Neural Fluid Surrogates](https://mlsanity.com/paper/2605.18816/) (2026)
- [Anatomy-Informed Neural Networks: Encoding Anatomic Priors in Loss and Architecture, with an SE(3) Formulation of Guidewire-Induced Aortoiliac Deformation](https://mlsanity.com/paper/2608.21332/) (2026)

## Abstract (arXiv)

Hemodynamic velocity fields in coronary arteries could be the basis of valuable biomarkers for diagnosis, prognosis and treatment planning in cardiovascular disease. Velocity fields are typically obtained from patient-specific 3D artery models via computational fluid dynamics (CFD). However, CFD simulation requires meticulous setup by experts and is time-intensive, which hinders large-scale acceptance in clinical practice. To address this, we propose graph neural networks (GNN) as an efficient black-box surrogate method to estimate 3D velocity fields mapped to the vertices of tetrahedral meshes of the artery lumen. We train these GNNs on synthetic artery models and CFD-based ground truth velocity fields. Once the GNN is trained, velocity estimates in a new and unseen artery can be obtained with 36-fold speed-up compared to CFD. We demonstrate how to construct an SE(3)-equivariant GNN that is independent of the spatial orientation of the input mesh and show how this reduces the necessary amount of training data compared to a baseline neural network.