# Universality of shallow and deep neural networks on non-Euclidean spaces

arXiv: https://arxiv.org/abs/2602.23381 · HTML: https://mlsanity.com/paper/2602.23381/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 31 references that are in the map.
- First posted on arXiv 3 Feb 2026.

## Builds on (most cited first, 14 of 14)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Deep, Skinny Neural Networks are not Universal Approximators](https://mlsanity.com/paper/1810.00393/) (2018)
- [Non-Euclidean Universal Approximation](https://mlsanity.com/paper/2006.02341/) (2020)
- [Approximation capability of two hidden layer feedforward neural networks with fixed weights](https://mlsanity.com/paper/2101.09181/) (2021)
- [On the approximation by single hidden layer feedforward neural networks with fixed weights](https://mlsanity.com/paper/1708.06219/) (2017)
- [Global universal approximation of functional input maps on weighted spaces](https://mlsanity.com/paper/2306.03303/) (2023)
- [On the Kolmogorov neural networks](https://mlsanity.com/paper/2311.00049/) (2023)
- [Mixtures of Neural Operators Reduce Active Complexity in Operator Learning](https://mlsanity.com/paper/2404.09101/) (2024)
- [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)
- [Universal approximation theorem for neural networks with inputs from a topological vector space](https://mlsanity.com/paper/2409.12913/) (2024)
- [Neural networks in non-metric spaces](https://mlsanity.com/paper/2406.09310/) (2024)
- [On shallow feedforward neural networks with inputs from a topological space](https://mlsanity.com/paper/2504.02321/) (2025)

## Abstract (arXiv)

We study shallow and deep neural networks whose inputs range over a general topological space. The model is built from a prescribed family of continuous feature maps and reduces to multilayer feedforward networks in the Euclidean case. We focus on the universal approximation property and establish general conditions under which such networks are dense in spaces of continuous vector-valued functions on arbitrary topological spaces and, in particular, locally convex spaces. Universality results obtained in the arbitrary-width case extend classical approximation theorems to non-Euclidean spaces. We also consider the deep narrow setting, in which the width of each hidden layer is uniformly bounded while the depth is allowed to grow. We identify conditions under which such networks retain the universal approximation property. As a concrete example, we employ Ostrand's extension of the Kolmogorov superposition theorem to derive an explicit universality result for products of compact metric spaces, with width bounds expressed in terms of topological dimension.