# Clifford-Steerable Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2402.14730 · HTML: https://mlsanity.com/paper/2402.14730/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 67 references that are in the map.
- First posted on arXiv 22 Feb 2024.

## Builds on (most cited first, 30 of 40)

- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [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)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Learning to Control PDEs with Differentiable Physics](https://mlsanity.com/paper/2001.07457/) (2020)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [CKConv: Continuous Kernel Convolution For Sequential Data](https://mlsanity.com/paper/2102.02611/) (2021)
- [Deep Scale-spaces: Equivariance Over Scale](https://mlsanity.com/paper/1905.11697/) (2019)
- [Scale-Equivariant Steerable Networks](https://mlsanity.com/paper/1910.11093/) (2019)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Roto-Translation Covariant Convolutional Networks for Medical Image Analysis](https://mlsanity.com/paper/1804.03393/) (2018)
- [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)
- [Clifford Group Equivariant Neural Networks](https://mlsanity.com/paper/2305.11141/) (2023)
- [Geometric Algebra Transformer](https://mlsanity.com/paper/2305.18415/) (2023)
- [A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels](https://mlsanity.com/paper/2010.10952/) (2020)
- [Group Equivariant Fourier Neural Operators for Partial Differential Equations](https://mlsanity.com/paper/2306.05697/) (2023)
- [Geometric Clifford Algebra Networks](https://mlsanity.com/paper/2302.06594/) (2023)

## Built on by (most cited first, 25 of 25)

- [Does equivariance matter at scale?](https://mlsanity.com/paper/2410.23179/) (2024)
- [Lorentz-Equivariant Geometric Algebra Transformers for High-Energy Physics](https://mlsanity.com/paper/2405.14806/) (2024)
- [Space-Time Continuous PDE Forecasting using Equivariant Neural Fields](https://mlsanity.com/paper/2406.06660/) (2024)
- [Platonic Transformers: A Solid Choice For Equivariance](https://mlsanity.com/paper/2510.03511/) (2025)
- [Controlled Generation with Equivariant Variational Flow Matching](https://mlsanity.com/paper/2506.18340/) (2025)
- [Multivector Neurons: Better and Faster O(n)-Equivariant Clifford Graph Neural Networks](https://mlsanity.com/paper/2406.04052/) (2024)
- [Learning Color Equivariant Representations](https://mlsanity.com/paper/2406.09588/) (2024)
- [TASI Lectures on Physics for Machine Learning](https://mlsanity.com/paper/2408.00082/) (2024)
- [Zatom-1: Towards a Multimodal Foundation Model for 3D Molecules and Materials](https://mlsanity.com/paper/2602.22251/) (2026)
- [Equivariant Neural Networks for General Linear Symmetries on Lie Algebras](https://mlsanity.com/paper/2510.22984/) (2025)
- [ILPO-NET: Network for the invariant recognition of arbitrary volumetric patterns in 3D](https://mlsanity.com/paper/2403.19612/) (2024)
- [Clifford Group Equivariant Diffusion Models for 3D Molecular Generation](https://mlsanity.com/paper/2504.15773/) (2025)
- [Surrogate Modeling of 3D Rayleigh-Benard Convection with Equivariant Autoencoders](https://mlsanity.com/paper/2505.13569/) (2025)
- [AdS-GNN -- a Conformally Equivariant Graph Neural Network](https://mlsanity.com/paper/2505.12880/) (2025)
- [GLGENN: A Novel Parameter-Light Equivariant Neural Networks Architecture Based on Clifford Geometric Algebras](https://mlsanity.com/paper/2506.09625/) (2025)
- [On the VC dimension of deep group convolutional neural networks](https://mlsanity.com/paper/2410.15800/) (2024)
- [Flow Equivariant Recurrent Neural Networks](https://mlsanity.com/paper/2507.14793/) (2025)
- [Adaptive Domain Models: Bayesian Evolution, Warm Rotation, and Principled Training for Geometric and Neuromorphic AI](https://mlsanity.com/paper/2603.18104/) (2026)
- [Geometric Algebra Meets Cartesian Tensors: Higher-Order Equivariance for Interatomic Potentials](https://mlsanity.com/paper/2606.29584/) (2026)
- [FoGE: Fock Space inspired encoding for graph prompting](https://mlsanity.com/paper/2507.02937/) (2025)
- [Conditional Clifford-Steerable CNNs for PDE Modeling](https://mlsanity.com/paper/2510.14007/) (2025)
- [On Lie Groups Preserving Subspaces of Degenerate Clifford Algebras](https://mlsanity.com/paper/2601.07191/) (2026)
- [Bases of Steerable Kernels for Equivariant CNNs: From 2D Rotations to the Lorentz Group](https://mlsanity.com/paper/2603.12459/) (2026)
- [Distillation of Foundation Models for Time-dependent PDEs](https://mlsanity.com/paper/2608.11937/) (2026)
- [Coordinate-Free Automatic Differentiation](https://mlsanity.com/paper/2610.12776/) (2026)

## Abstract (arXiv)

We present Clifford-Steerable Convolutional Neural Networks (CS-CNNs), a novel class of E(p, q)-equivariant CNNs. CS-CNNs process multivector fields on pseudo-Euclidean spaces ℝp,q. They cover, for instance, E(3)-equivariance on ℝ3 and Poincar'e-equivariance on Minkowski spacetime ℝ1,3. Our approach is based on an implicit parametrization of O(p,q)-steerable kernels via Clifford group equivariant neural networks. We significantly and consistently outperform baseline methods on fluid dynamics as well as relativistic electrodynamics forecasting tasks.