# Geometric Clifford Algebra Networks

arXiv: https://arxiv.org/abs/2302.06594 · HTML: https://mlsanity.com/paper/2302.06594/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- Builds on 74 of its 130 references that are in the map.
- First posted on arXiv 13 Feb 2023.

## Builds on (most cited first, 30 of 74)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [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)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Empirical Evaluation of Rectified Activations in Convolutional Network](https://mlsanity.com/paper/1505.00853/) (2015)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)

## Built on by (most cited first, 30 of 38)

- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Clifford Group Equivariant Neural Networks](https://mlsanity.com/paper/2305.11141/) (2023)
- [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)
- [Lorentz-Equivariant Geometric Algebra Transformers for High-Energy Physics](https://mlsanity.com/paper/2405.14806/) (2024)
- [Clifford-Steerable Convolutional Neural Networks](https://mlsanity.com/paper/2402.14730/) (2024)
- [Clifford Group Equivariant Simplicial Message Passing Networks](https://mlsanity.com/paper/2402.10011/) (2024)
- [AI Foundation Models for Weather and Climate: Applications, Design, and Implementation](https://mlsanity.com/paper/2309.10808/) (2023)
- [MatSciML: A Broad, Multi-Task Benchmark for Solid-State Materials Modeling](https://mlsanity.com/paper/2309.05934/) (2023)
- [Learning Lagrangian Fluid Mechanics with E(3)-Equivariant Graph Neural Networks](https://mlsanity.com/paper/2305.15603/) (2023)
- [Fast Dynamic 1D Simulation of Divertor Plasmas with Neural PDE Surrogates](https://mlsanity.com/paper/2305.18944/) (2023)
- [GAGrasp: Geometric Algebra Diffusion for Dexterous Grasping](https://mlsanity.com/paper/2503.04123/) (2025)
- [Generating Highly Designable Proteins with Geometric Algebra Flow Matching](https://mlsanity.com/paper/2411.05238/) (2024)
- [Differentiable and Learnable Wireless Simulation with Geometric Transformers](https://mlsanity.com/paper/2406.14995/) (2024)
- [Multivector Neurons: Better and Faster O(n)-Equivariant Clifford Graph Neural Networks](https://mlsanity.com/paper/2406.04052/) (2024)
- [Understanding Vector-Valued Neural Networks and Their Relationship with Real and Hypercomplex-Valued Neural Networks](https://mlsanity.com/paper/2309.07716/) (2023)
- [Clifford Group Equivariant Diffusion Models for 3D Molecular Generation](https://mlsanity.com/paper/2504.15773/) (2025)
- [Hybrid Diffusion Policies with Projective Geometric Algebra for Efficient Robot Manipulation Learning](https://mlsanity.com/paper/2507.05695/) (2025)
- [Ananke: Contractive Torus Attractor Networks](https://mlsanity.com/paper/2609.24737/) (2026)
- [Randomized Geometric Algebra Methods for Convex Neural Networks](https://mlsanity.com/paper/2406.02806/) (2024)
- [Fully tensorial approach to hypercomplex-valued neural networks](https://mlsanity.com/paper/2407.00449/) (2024)
- [Geometric Algebra Meets Large Language Models: Instruction-Based Transformations of Separate Meshes in 3D, Interactive and Controllable Scenes](https://mlsanity.com/paper/2408.02275/) (2024)
- [Geometric Algebra Meets Cartesian Tensors: Higher-Order Equivariance for Interatomic Potentials](https://mlsanity.com/paper/2606.29584/) (2026)
- [Compact Matrix Quantum Group Equivariant Neural Networks](https://mlsanity.com/paper/2311.06358/) (2023)
- [STAResNet: a Network in Spacetime Algebra to solve Maxwell's PDEs](https://mlsanity.com/paper/2408.13619/) (2024)
- [Fast Clifford Neural Layers](https://mlsanity.com/paper/2507.01040/) (2025)
- [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)
- [Geometric Dynamics of Consumer Credit Cycles: A Multivector-based Linear-Attention Framework for Explanatory Economic Analysis](https://mlsanity.com/paper/2510.15892/) (2025)

## Abstract (arXiv)

We propose Geometric Clifford Algebra Networks (GCANs) for modeling dynamical systems. GCANs are based on symmetry group transformations using geometric (Clifford) algebras. We first review the quintessence of modern (plane-based) geometric algebra, which builds on isometries encoded as elements of the Pin(p,q,r) group. We then propose the concept of group action layers, which linearly combine object transformations using pre-specified group actions. Together with a new activation and normalization scheme, these layers serve as adjustable textitgeometric templates that can be refined via gradient descent. Theoretical advantages are strongly reflected in the modeling of three-dimensional rigid body transformations as well as large-scale fluid dynamics simulations, showing significantly improved performance over traditional methods.