# On the Kolmogorov neural networks

arXiv: https://arxiv.org/abs/2311.00049 · HTML: https://mlsanity.com/paper/2311.00049/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 36 references that are in the map.
- First posted on arXiv 31 Oct 2023.

## Builds on (most cited first, 4 of 4)

- [The Kolmogorov-Arnold representation theorem revisited](https://mlsanity.com/paper/2007.15884/) (2020)
- [Neural Network Approximation: Three Hidden Layers Are Enough](https://mlsanity.com/paper/2010.14075/) (2020)
- [Error bounds for deep ReLU networks using the Kolmogorov--Arnold superposition theorem](https://mlsanity.com/paper/1906.11945/) (2019)
- [Approximation capability of two hidden layer feedforward neural networks with fixed weights](https://mlsanity.com/paper/2101.09181/) (2021)

## Built on by (most cited first, 19 of 19)

- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Demonstrating the Efficacy of Kolmogorov-Arnold Networks in Vision Tasks](https://mlsanity.com/paper/2406.14916/) (2024)
- [A Practitioner's Guide to Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2510.25781/) (2025)
- [LeanKAN: A Parameter-Lean Kolmogorov-Arnold Network Layer with Improved Memory Efficiency and Convergence Behavior](https://mlsanity.com/paper/2502.17844/) (2025)
- [Universal Approximation Theorem for Vector- and Hypercomplex-Valued Neural Networks](https://mlsanity.com/paper/2401.02277/) (2024)
- [Addressing common misinterpretations of KART and UAT in neural network literature](https://mlsanity.com/paper/2408.16389/) (2024)
- [Acceleration of Grokking in Learning Arithmetic Operations via Kolmogorov-Arnold Representation](https://mlsanity.com/paper/2405.16658/) (2024)
- [Structured and Balanced Multi-Component and Multi-Layer Neural Networks](https://mlsanity.com/paper/2407.00765/) (2024)
- [Physics-informed KAN PointNet: Deep learning for simultaneous solutions to inverse problems in incompressible flow on numerous irregular geometries](https://mlsanity.com/paper/2504.06327/) (2025)
- [Relating Piecewise Linear Kolmogorov Arnold Networks to ReLU Networks](https://mlsanity.com/paper/2503.01702/) (2025)
- [On shallow feedforward neural networks with inputs from a topological space](https://mlsanity.com/paper/2504.02321/) (2025)
- [Generalizable autoregressive modeling of time series through functional narratives](https://mlsanity.com/paper/2410.08421/) (2024)
- [GrokFormer: Graph Fourier Kolmogorov-Arnold Transformers](https://mlsanity.com/paper/2411.17296/) (2024)
- [Kolmogorov GAM Networks are all you need!](https://mlsanity.com/paper/2501.00704/) (2025)
- [Exploring Kolmogorov-Arnold Network Expansions in Vision Transformers for Mitigating Catastrophic Forgetting in Continual Learning](https://mlsanity.com/paper/2507.04020/) (2025)
- [The Function-Representation Model of Computation](https://mlsanity.com/paper/2410.07928/) (2024)
- [Clifford Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2602.05977/) (2026)
- [Universality of shallow and deep neural networks on non-Euclidean spaces](https://mlsanity.com/paper/2602.23381/) (2026)
- [Kolmogorov--Arnold stability for discontinuous functions](https://mlsanity.com/paper/2609.07240/) (2026)

## Abstract (arXiv)

In this paper, we show that the Kolmogorov two hidden layer neural network model with a continuous, discontinuous bounded or unbounded activation function in the second hidden layer can precisely represent continuous, discontinuous bounded and all unbounded multivariate functions, respectively.