SO(3)-Equivariant CNN on HEALPix Meshes: Learning Without Relearning Physical Symmetry

Forum topic · 小凯 · 2026-10-20

Summary

A forum post discusses an arXiv paper (2610.23821) by Joshua Williamson, Niall Jeffrey, and Lorne Whiteway introducing a spherical gauge equivariant mesh architecture designed for HEALPix pixelization. Many key physical science datasets live on the sphere with no preferred position or orientation; methods ignoring this symmetry demand far more simulations, which can bias scientific analyses. The proposed SO(3)-equivariant convolution is built from steerable filters, parallel transport, and spin-weighted features, providing exact symmetry by construction so the network need not learn it from training data. Convolutions are provably exactly equivariant at a given band limit. The full CNN achieves percent-level or lower SO(3)-invariance error controlled by well-understood hyperparameters. On benchmarks, the method achieves top accuracy in dense gradient regression and is competitive on CMB foreground removal and denoising. Benefits are largest for simulation-based inference (SBI), where high-fidelity simulations are expensive: in cosmological lensing SBI, the method yields the tightest posteriors at a fixed simulation budget.

Paper Overview

Field: Machine Learning Authors: Joshua Williamson, Niall Jeffrey, Lorne Whiteway Published: 2026-10-16 arXiv: 2610.23821

Abstract

Many of the most important datasets in the physical sciences live natively on the sphere, and these data often carry no preferred position or orientation. Using methods that do not reflect this symmetry leads to far higher simulation requirements, which can be catastrophic for scientific analyses (forcing reliance on cheap, imperfect models that bias results). The authors present a novel spherical implementation of a gauge equivariant mesh architecture specifically designed for the Hierarchical Equal Area isoLatitude Pixelization (HEALPix) of the sphere.

The SO(3)-equivariant convolution is built from steerable filters, parallel transport, and spin-weighted features. The architecture supplies the exact symmetry, so the network does not spend training data learning it. The convolution operations are constructed to provide provably exact equivariance at a given band limit.

Key Results

The full CNN achieves percent-level and lower SO(3)-invariance error, controllable through well-understood hyperparameters.
On benchmarks, the technique achieves the highest accuracy on dense gradient regression and is competitive on cosmic microwave background (CMB) foreground removal and denoising.
Gains in simulation efficiency matter most for simulation-based inference (SBI): high-fidelity simulations are expensive, neural compressors require large training sets, and fixed budgets force reliance on less accurate simulators.
In cosmological lensing SBI, the method produces the tightest posteriors given the same number of simulations.

Source

Paper: arXiv:2610.23821

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