# Universal approximation results for neural networks with non-polynomial activation function over non-compact domains

arXiv: https://arxiv.org/abs/2410.14759 · HTML: https://mlsanity.com/paper/2410.14759/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 62 references that are in the map.
- First posted on arXiv 18 Oct 2024.

## Builds on (most cited first, 13 of 13)

- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Optimal Approximation with Sparsely Connected Deep Neural Networks](https://mlsanity.com/paper/1705.01714/) (2017)
- [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021)
- [Neural Network with Unbounded Activation Functions is Universal Approximator](https://mlsanity.com/paper/1505.03654/) (2015)
- [Universal Approximation Theorems for Differentiable Geometric Deep Learning](https://mlsanity.com/paper/2101.05390/) (2021)
- [Deep Network Approximation: Beyond ReLU to Diverse Activation Functions](https://mlsanity.com/paper/2307.06555/) (2023)
- [Risk Bounds for High-dimensional Ridge Function Combinations Including Neural Networks](https://mlsanity.com/paper/1607.01434/) (2016)
- [Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer](https://mlsanity.com/paper/2201.13094/) (2022)
- [Differentiable reservoir computing](https://mlsanity.com/paper/1902.06094/) (2019)
- [Global universal approximation of functional input maps on weighted spaces](https://mlsanity.com/paper/2306.03303/) (2023)
- [Rates of Approximation by ReLU Shallow Neural Networks](https://mlsanity.com/paper/2307.12461/) (2023)
- [Uniform error estimates for artificial neural network approximations for heat equations](https://mlsanity.com/paper/1911.09647/) (2019)
- [Noncompact uniform universal approximation](https://mlsanity.com/paper/2308.03812/) (2023)

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

- [Solving stochastic partial differential equations using neural networks in the Wiener chaos expansion](https://mlsanity.com/paper/2411.03384/) (2024)
- [Any-Dimensional Invariant Universality](https://mlsanity.com/paper/2605.23156/) (2026)
- [Distributionally Robust Deep Q-Learning](https://mlsanity.com/paper/2505.19058/) (2025)
- [Neural Measures for learning distributions of Random PDEs](https://mlsanity.com/paper/2507.01687/) (2025)
- [Approximation rates of quantum neural networks for periodic functions via Jackson's inequality](https://mlsanity.com/paper/2511.16149/) (2025)
- [Distributionally robust approximation property of neural networks](https://mlsanity.com/paper/2510.09177/) (2025)
- [Neural operators approximate strongly continuous convex monotone semigroups](https://mlsanity.com/paper/2609.02727/) (2026)

## Abstract (arXiv)

This paper extends the universal approximation property of single-hidden-layer feedforward neural networks beyond compact domains, which is of particular interest for the approximation within weighted Ck-spaces and weighted Sobolev spaces over unbounded domains. More precisely, by assuming that the activation function is non-polynomial, we establish universal approximation results within function spaces defined over non-compact subsets of a Euclidean space, including Lp-spaces, weighted Ck-spaces, and weighted Sobolev spaces, where the latter two include the approximation of the (weak) derivatives. Moreover, we provide some dimension-independent rates for approximating a function with sufficiently regular and integrable Fourier transform by neural networks with non-polynomial activation function.