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This study demonstrates that explicit $E(3)$-equivariance in neural fluid surrogates consistently outperforms non-equivariant models and rotational augmentation across hemodynamic benchmarks with natural orientation variation. The research introduces the Anchored-Branched Geometric Algebra Transformer (AB-GATr), which achieves competitive accuracy with significantly lower training costs compared to existing equivariant baselines. These findings clarify that while canonical alignment can occasionally favor simpler architectures, robust performance under broad rotation distributions strongly supports the use of geometric inductive biases in computational fluid dynamics.
When Does Equivariance Help? Canonical Alignment in Neural Fluid Surrogates
arXiv:2605.18816v2 Announce Type: replace Abstract: 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.
Title:When Does Equivariance Help? Canonical Alignment in Neural Fluid Surrogates
Abstract: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.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2605.18816 [cs.LG]
(or arXiv:2605.18816v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2605.18816 Focus to learn more arXiv-issued DOI via DataCite
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