# Geometric Kolmogorov-Arnold Superposition Theorem

arXiv: https://arxiv.org/abs/2502.16664 · HTML: https://mlsanity.com/paper/2502.16664/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 71 references that are in the map.
- First posted on arXiv 23 Feb 2025.

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

- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [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)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [Machine Learning Force Fields](https://mlsanity.com/paper/2010.07067/) (2020)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [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)
- [Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximation](https://mlsanity.com/paper/2405.07200/) (2024)
- [A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks](https://mlsanity.com/paper/2406.02917/) (2024)
- [Energy Flow Networks: Deep Sets for Particle Jets](https://mlsanity.com/paper/1810.05165/) (2018)
- [DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for Mechanics Problems](https://mlsanity.com/paper/2405.19143/) (2024)
- [Kolmogorov-Arnold Networks for Time Series: Bridging Predictive Power and Interpretability](https://mlsanity.com/paper/2406.02496/) (2024)
- [A Survey on Kolmogorov-Arnold Network](https://mlsanity.com/paper/2411.06078/) (2024)
- [Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks](https://mlsanity.com/paper/2406.11045/) (2024)
- [Kolmogorov-Arnold Transformer](https://mlsanity.com/paper/2409.10594/) (2024)
- [Generative Adversarial Symmetry Discovery](https://mlsanity.com/paper/2302.00236/) (2023)
- [Enhancing Graph Collaborative Filtering with FourierKAN Feature Transformation](https://mlsanity.com/paper/2406.01034/) (2024)
- [Neural Injective Functions for Multisets, Measures and Graphs via a Finite Witness Theorem](https://mlsanity.com/paper/2306.06529/) (2023)
- [Kolmogorov-Arnold Networks: A Critical Assessment of Claims, Performance, and Practical Viability](https://mlsanity.com/paper/2407.11075/) (2024)
- [Higher-Rank Irreducible Cartesian Tensors for Equivariant Message Passing](https://mlsanity.com/paper/2405.14253/) (2024)
- [Deep Learning Alternatives of the Kolmogorov Superposition Theorem](https://mlsanity.com/paper/2410.01990/) (2024)
- [KANICE: Kolmogorov-Arnold Networks with Interactive Convolutional Elements](https://mlsanity.com/paper/2410.17172/) (2024)
- [EPi-cKANs: Elasto-Plasticity Informed Kolmogorov-Arnold Networks Using Chebyshev Polynomials](https://mlsanity.com/paper/2410.10897/) (2024)

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

- [Variational Kolmogorov-Arnold Network](https://mlsanity.com/paper/2507.02466/) (2025)
- [Clifford Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2602.05977/) (2026)

## Abstract (arXiv)

The Kolmogorov-Arnold Theorem (KAT), or more generally, the Kolmogorov Superposition Theorem (KST), establishes that any non-linear multivariate function can be exactly represented as a finite superposition of non-linear univariate functions. Unlike the universal approximation theorem, which provides only an approximate representation without guaranteeing a fixed network size, KST offers a theoretically exact decomposition. The Kolmogorov-Arnold Network (KAN) was introduced as a trainable model to implement KAT, and recent advancements have adapted KAN using concepts from modern neural networks. However, KAN struggles to effectively model physical systems that require inherent equivariance or invariance geometric symmetries as E(3) transformations, a key property for many scientific and engineering applications. In this work, we propose a novel extension of KAT and KAN to incorporate equivariance and invariance over various group actions, including O(n), O(1,n), Sn, and general GL, enabling accurate and efficient modeling of these systems. Our approach provides a unified approach that bridges the gap between mathematical theory and practical architectures for physical systems, expanding the applicability of KAN to a broader class of problems. We provide experimental validation on molecular dynamical systems and particle physics.