# SO(3)-Equivariant Neural Networks for Learning from Scalar and Vector Fields on Spheres

arXiv: https://arxiv.org/abs/2503.09456 · HTML: https://mlsanity.com/paper/2503.09456/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 38 references that are in the map.
- First posted on arXiv 12 Mar 2025.

## Builds on (most cited first, 22 of 22)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [GenCast: Diffusion-based ensemble forecasting for medium-range weather](https://mlsanity.com/paper/2312.15796/) (2023)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Polar Transformer Networks](https://mlsanity.com/paper/1709.01889/) (2017)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Spin-Weighted Spherical CNNs](https://mlsanity.com/paper/2006.10731/) (2020)
- [3D-Rotation-Equivariant Quaternion Neural Networks](https://mlsanity.com/paper/1911.09040/) (2019)
- [Equivariance versus Augmentation for Spherical Images](https://mlsanity.com/paper/2202.03990/) (2022)
- [Homogeneous vector bundles and G-equivariant convolutional neural networks](https://mlsanity.com/paper/2105.05400/) (2021)
- [Scaling Spherical CNNs](https://mlsanity.com/paper/2306.05420/) (2023)
- [HEAL-SWIN: A Vision Transformer On The Sphere](https://mlsanity.com/paper/2307.07313/) (2023)

## Built on by (most cited first, 2 of 2)

- [Encoding the Euler Characteristic Transform](https://mlsanity.com/paper/2606.10824/) (2026)
- [Physically Typed and Geometry-Aware Representations for Earth Foundation Models](https://mlsanity.com/paper/2609.13868/) (2026)

## Abstract (arXiv)

Analyzing scalar and vector fields on the sphere, such as temperature or wind speed and direction on Earth, is a difficult task. Models should respect both the rotational symmetries of the sphere and the inherent symmetries of the vector fields. A class of equivariant models has emerged, which process these spherical signals by applying group convolutions in Fourier space with respect to the three-dimensional rotation group. However, the proposed models are constrained in the choice of convolution kernels and nonlinearities in order to preserve the desired signal properties. In this paper, we introduce a deep learning architecture without these limitations, thus with a richer class of convolution kernels and activation functions. This architecture is suitable for signals consisting of both scalar and vector fields on the sphere, as they can be described as equivariant signals on the three-dimensional rotation group. Experiments show that this architecture generally outperforms standard CNNs and often matches or exceeds the performance of spherical CNNs trained under comparable conditions. However, the advantage over sCNNs is not uniform across all tasks and we observe that incorporating the interaction between different spins in the hidden layers narrows this gap.