# Any-dimensional equivariant neural networks

arXiv: https://arxiv.org/abs/2306.06327 · HTML: https://mlsanity.com/paper/2306.06327/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 61 references that are in the map.
- First posted on arXiv 10 Jun 2023.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [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)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (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)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Building powerful and equivariant graph neural networks with structural message-passing](https://mlsanity.com/paper/2006.15107/) (2020)
- [What is an equivariant neural network?](https://mlsanity.com/paper/2205.07362/) (2022)
- [Dimensionless machine learning: Imposing exact units equivariance](https://mlsanity.com/paper/2204.00887/) (2022)

## Built on by (most cited first, 9 of 9)

- [Symmetry-Based Structured Matrices for Efficient Approximately Equivariant Networks](https://mlsanity.com/paper/2409.11772/) (2024)
- [Graph neural networks and non-commuting operators](https://mlsanity.com/paper/2411.04265/) (2024)
- [Any-Dimensional Invariant Universality](https://mlsanity.com/paper/2605.23156/) (2026)
- [From Learning to Optimize to Learning Optimization Algorithms](https://mlsanity.com/paper/2405.18222/) (2024)
- [Towards Coordinate- and Dimension-Agnostic Machine Learning for Partial Differential Equations](https://mlsanity.com/paper/2505.16549/) (2025)
- [Axial Neural Networks for Dimension-Free Foundation Models](https://mlsanity.com/paper/2510.13665/) (2025)
- [Graded Representation Theory of Equivariant Neural Networks](https://mlsanity.com/paper/2609.25776/) (2026)
- [Warm-starting PDE solvers with any-dimensional machine learning](https://mlsanity.com/paper/2609.38916/) (2026)
- [Dimension-Free Scaling Laws for Size Generalization: An Exponential Separation](https://mlsanity.com/paper/2610.19279/) (2026)

## Abstract (arXiv)

Traditional supervised learning aims to learn an unknown mapping by fitting a function to a set of input-output pairs with a fixed dimension. The fitted function is then defined on inputs of the same dimension. However, in many settings, the unknown mapping takes inputs in any dimension; examples include graph parameters defined on graphs of any size and physics quantities defined on an arbitrary number of particles. We leverage a newly-discovered phenomenon in algebraic topology, called representation stability, to define equivariant neural networks that can be trained with data in a fixed dimension and then extended to accept inputs in any dimension. Our approach is user-friendly, requiring only the network architecture and the groups for equivariance, and can be combined with any training procedure. We provide a simple open-source implementation of our methods and offer preliminary numerical experiments.