# When Does Equivariance Help? Canonical Alignment in Neural Fluid Surrogates

arXiv: https://arxiv.org/abs/2605.18816 · HTML: https://mlsanity.com/paper/2605.18816/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 49 references that are in the map.
- First posted on arXiv 12 May 2026.

## Builds on (most cited first, 30 of 32)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Perceiver IO: A General Architecture for Structured Inputs & Outputs](https://mlsanity.com/paper/2107.14795/) (2021)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Transolver++: An Accurate Neural Solver for PDEs on Million-Scale Geometries](https://mlsanity.com/paper/2502.02414/) (2025)
- [Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs](https://mlsanity.com/paper/2003.05425/) (2020)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Swallowing the Bitter Pill: Simplified Scalable Conformer Generation](https://mlsanity.com/paper/2311.17932/) (2023)
- [AB-UPT: Scaling Neural CFD Surrogates for High-Fidelity Automotive Aerodynamics Simulations via Anchored-Branched Universal Physics Transformers](https://mlsanity.com/paper/2502.09692/) (2025)
- [Does equivariance matter at scale?](https://mlsanity.com/paper/2410.23179/) (2024)
- [Geometric Algebra Transformer](https://mlsanity.com/paper/2305.18415/) (2023)
- [Fast, Expressive SE(n) Equivariant Networks through Weight-Sharing in Position-Orientation Space](https://mlsanity.com/paper/2310.02970/) (2023)
- [Mesh Neural Networks for SE(3)-Equivariant Hemodynamics Estimation on the Artery Wall](https://mlsanity.com/paper/2212.05023/) (2022)
- [Erwin: A Tree-based Hierarchical Transformer for Large-scale Physical Systems](https://mlsanity.com/paper/2502.17019/) (2025)
- [Grounding Continuous Representations in Geometry: Equivariant Neural Fields](https://mlsanity.com/paper/2406.05753/) (2024)
- [On genuine invariance learning without weight-tying](https://mlsanity.com/paper/2308.03904/) (2023)
- [LaB-GATr: geometric algebra transformers for large biomedical surface and volume meshes](https://mlsanity.com/paper/2403.07536/) (2024)
- [Probing Equivariance and Symmetry Breaking in Convolutional Networks](https://mlsanity.com/paper/2501.01999/) (2025)
- [SE(3) symmetry lets graph neural networks learn arterial velocity estimation from small datasets](https://mlsanity.com/paper/2302.08780/) (2023)
- [Platonic Transformers: A Solid Choice For Equivariance](https://mlsanity.com/paper/2510.03511/) (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)
- [Geometry aware inference of steady state PDEs using Equivariant Neural Fields representations](https://mlsanity.com/paper/2504.18591/) (2025)

## Abstract (arXiv)

Neural surrogates can accelerate computational fluid dynamics (CFD) simulations by orders of magnitude, but practical deployment in engineering and healthcare applications requires architectures that scale to high-resolution meshes and learn effectively from limited data. Explicit equivariance offers a principled inductive bias, yet its accuracy benefits may depend on the prediction task and the distribution of anatomical orientations. We investigate this dependence across three hemodynamic benchmarks with different degrees of natural canonical alignment. To support this study, we introduce the Anchored-Branched Geometric Algebra Transformer (AB-GATr), an E(3)-equivariant surrogate that efficiently predicts coupled surface and volume quantities. Across these benchmarks, AB-GATr consistently outperforms the evaluated non-equivariant models, including variants trained with rotational augmentation, while achieving accuracy competitive with E(3)-equivariant LaB-GATr at substantially lower training cost. In comparison, rotational augmentation provides inconsistent benefits across architectures and can reduce accuracy. A controlled experiment on ShapeNet-Car shows that strong canonical alignment can favor non-equivariant models, but their accuracy generally deteriorates as training orientations broaden and can decline sharply under broader test rotations. We further investigate these patterns using extended symmetry-breaking diagnostics and probes of the predictive information associated with canonical alignment across all benchmarks. Together, these results support explicit equivariance for the evaluated hemodynamic tasks with natural orientation variation, while showing that its accuracy benefits depend on the task and orientation distribution.