# Universal Approximation Theorem for Neural Networks

arXiv: https://arxiv.org/abs/2102.10993 · HTML: https://mlsanity.com/paper/2102.10993/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 27 references that are in the map.
- First posted on arXiv 19 Feb 2021.

## Builds on (most cited first, 3 of 3)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Neural Network with Unbounded Activation Functions is Universal Approximator](https://mlsanity.com/paper/1505.03654/) (2015)
- [The Barron Space and the Flow-induced Function Spaces for Neural Network Models](https://mlsanity.com/paper/1906.08039/) (2019)

## Built on by (most cited first, 7 of 7)

- [fKAN: Fractional Kolmogorov-Arnold Networks with trainable Jacobi basis functions](https://mlsanity.com/paper/2406.07456/) (2024)
- [A case study of spatiotemporal forecasting techniques for weather forecasting](https://mlsanity.com/paper/2209.14782/) (2022)
- [Neuronal Attention Circuit (NAC) for Representation Learning](https://mlsanity.com/paper/2512.10282/) (2025)
- [How important are specialized transforms in Neural Operators?](https://mlsanity.com/paper/2308.09293/) (2023)
- [Robust Noisy Label Learning via Two-Stream Sample Distillation](https://mlsanity.com/paper/2404.10499/) (2024)
- [LSTM Autoencoder-based Deep Neural Networks for Barley Genotype-to-Phenotype Prediction](https://mlsanity.com/paper/2407.16709/) (2024)
- [An End-to-End PyTorch Interface for Differentiable PDE Solvers: A RANS Model-Correction Study](https://mlsanity.com/paper/2605.28858/) (2026)

## Abstract (arXiv)

Is there any theoretical guarantee for the approximation ability of neural networks? The answer to this question is the "Universal Approximation Theorem for Neural Networks". This theorem states that a neural network is dense in a certain function space under an appropriate setting. This paper is a comprehensive explanation of the universal approximation theorem for feedforward neural networks, its approximation rate problem (the relation between the number of intermediate units and the approximation error), and Barron space in Japanese.