# Mesh Neural Networks for SE(3)-Equivariant Hemodynamics Estimation on the Artery Wall

arXiv: https://arxiv.org/abs/2212.05023 · HTML: https://mlsanity.com/paper/2212.05023/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 59 references that are in the map.
- First posted on arXiv 9 Dec 2022.

## Builds on (most cited first, 14 of 14)

- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Deep Learning for 3D Point Clouds: A Survey](https://mlsanity.com/paper/1912.12033/) (2019)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [DiffusionNet: Discretization Agnostic Learning on Surfaces](https://mlsanity.com/paper/2012.00888/) (2020)
- [Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems](https://mlsanity.com/paper/2107.12146/) (2021)
- [Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs](https://mlsanity.com/paper/2003.05425/) (2020)
- [MeshCNN: A Network with an Edge](https://mlsanity.com/paper/1809.05910/) (2018)
- [CNNs on Surfaces using Rotation-Equivariant Features](https://mlsanity.com/paper/2006.01570/) (2020)
- [The Vector Heat Method](https://mlsanity.com/paper/1805.09170/) (2018)
- [Scalable algorithms for physics-informed neural and graph networks](https://mlsanity.com/paper/2205.08332/) (2022)
- [Graph Convolutional Networks for Coronary Artery Segmentation in Cardiac CT Angiography](https://mlsanity.com/paper/1908.05343/) (2019)
- [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, 29 of 29)

- [Does equivariance matter at scale?](https://mlsanity.com/paper/2410.23179/) (2024)
- [Geometric Algebra Transformer](https://mlsanity.com/paper/2305.18415/) (2023)
- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [Euclidean, Projective, Conformal: Choosing a Geometric Algebra for Equivariant Transformers](https://mlsanity.com/paper/2311.04744/) (2023)
- [LaB-GATr: geometric algebra transformers for large biomedical surface and volume meshes](https://mlsanity.com/paper/2403.07536/) (2024)
- [Machine learning for modelling unstructured grid data in computational physics: a review](https://mlsanity.com/paper/2502.09346/) (2025)
- [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)
- [Training Transformers for Mesh-Based Simulations](https://mlsanity.com/paper/2508.18051/) (2025)
- [Modeling Dynamics over Meshes with Gauge Equivariant Nonlinear Message Passing](https://mlsanity.com/paper/2310.19589/) (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)
- [Graph Deep Learning for Intracranial Aneurysm Blood Flow Simulation and Risk Assessment](https://mlsanity.com/paper/2512.09013/) (2025)
- [SIRE: scale-invariant, rotation-equivariant estimation of artery orientations using graph neural networks](https://mlsanity.com/paper/2311.05400/) (2023)
- [Universal Collection of Euclidean Invariants between Pairs of Position-Orientations](https://mlsanity.com/paper/2504.03299/) (2025)
- [Wall Shear Stress Estimation in Abdominal Aortic Aneurysms: Towards Generalisable Neural Surrogate Models](https://mlsanity.com/paper/2507.22817/) (2025)
- [GReAT: leveraging geometric artery data to improve wall shear stress assessment](https://mlsanity.com/paper/2508.19030/) (2025)
- [TRELLIS-Enhanced Surface Features for Comprehensive Intracranial Aneurysm Analysis](https://mlsanity.com/paper/2509.03095/) (2025)
- [Mesh-Informed Reduced Order Models for Aneurysm Rupture Risk Prediction](https://mlsanity.com/paper/2410.03802/) (2024)
- [Permutation Equivariant Neural Networks for Symmetric Tensors](https://mlsanity.com/paper/2503.11276/) (2025)
- [Implicit Augmentation from Distributional Symmetry in Turbulence Super-Resolution](https://mlsanity.com/paper/2509.20683/) (2025)
- [Physics-informed self-supervised learning for predictive modeling of coronary artery digital twins](https://mlsanity.com/paper/2512.03055/) (2025)
- [Latent Dynamics Graph Convolutional Networks for model order reduction of parameterized time-dependent PDEs](https://mlsanity.com/paper/2601.11259/) (2026)
- [Smoothness Errors in Dynamics Models and How to Avoid Them](https://mlsanity.com/paper/2602.05352/) (2026)
- [Stable Long-Horizon Spatiotemporal Prediction on Meshes Using Latent Multiscale Recurrent Graph Neural Networks](https://mlsanity.com/paper/2602.18146/) (2026)
- [Mesh Based Simulations with Spatial and Temporal awareness](https://mlsanity.com/paper/2605.01542/) (2026)
- [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)
- [Learning PDE Dynamics between Submanifolds Using Green's Observation Operators](https://mlsanity.com/paper/2610.01697/) (2026)

## Abstract (arXiv)

Computational fluid dynamics (CFD) is a valuable asset for patient-specific cardiovascular-disease diagnosis and prognosis, but its high computational demands hamper its adoption in practice. Machine-learning methods that estimate blood flow in individual patients could accelerate or replace CFD simulation to overcome these limitations. In this work, we consider the estimation of vector-valued quantities on the wall of three-dimensional geometric artery models. We employ group equivariant graph convolution in an end-to-end SE(3)-equivariant neural network that operates directly on triangular surface meshes and makes efficient use of training data. We run experiments on a large dataset of synthetic coronary arteries and find that our method estimates directional wall shear stress (WSS) with an approximation error of 7.6% and normalised mean absolute error (NMAE) of 0.4% while up to two orders of magnitude faster than CFD. Furthermore, we show that our method is powerful enough to accurately predict transient, vector-valued WSS over the cardiac cycle while conditioned on a range of different inflow boundary conditions. These results demonstrate the potential of our proposed method as a plugin replacement for CFD in the personalised prediction of hemodynamic vector and scalar fields.