# Approximation capabilities of neural networks on unbounded domains

arXiv: https://arxiv.org/abs/1910.09293 · HTML: https://mlsanity.com/paper/1910.09293/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Oct 2019.

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

- [Minimum width for universal approximation using ReLU networks on compact domain](https://mlsanity.com/paper/2309.10402/) (2023)
- [Universal approximation and model compression for radial neural networks](https://mlsanity.com/paper/2107.02550/) (2021)
- [Weighted Sobolev Approximation Rates for Neural Networks on Unbounded Domains](https://mlsanity.com/paper/2411.04108/) (2024)
- [Universal Time-Uniform Trajectory Approximation for Random Dynamical Systems with Recurrent Neural Networks](https://mlsanity.com/paper/2211.08018/) (2022)
- [Data Topology-Dependent Upper Bounds of Neural Network Widths](https://mlsanity.com/paper/2305.16375/) (2023)
- [Neural Networks Trained by Weight Permutation are Universal Approximators](https://mlsanity.com/paper/2407.01033/) (2024)
- [LUNA: Linear Universal Neural Attention with Generalization Guarantees](https://mlsanity.com/paper/2512.08061/) (2025)

## Abstract (arXiv)

In this paper, we prove that a shallow neural network with a monotone sigmoid, ReLU, ELU, Softplus, or LeakyReLU activation function can arbitrarily well approximate any L^p(p>=2) integrable functions defined on R*[0,1]^n. We also prove that a shallow neural network with a sigmoid, ReLU, ELU, Softplus, or LeakyReLU activation function expresses no nonzero integrable function defined on the Euclidean plane. Together with a recent result that the deep ReLU network can arbitrarily well approximate any integrable function on Euclidean spaces, we provide a new perspective on the advantage of multiple hidden layers in the context of ReLU networks. Lastly, we prove that the ReLU network with depth 3 is a universal approximator in L^p(R^n).