# Universal approximation theorem for neural networks with inputs from a topological vector space

arXiv: https://arxiv.org/abs/2409.12913 · HTML: https://mlsanity.com/paper/2409.12913/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 24 references that are in the map.
- First posted on arXiv 19 Sep 2024.

## Builds on (most cited first, 5 of 5)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Two-layer neural networks with values in a Banach space](https://mlsanity.com/paper/2105.02095/) (2021)
- [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)
- [Universal Approximation Theorem for Vector- and Hypercomplex-Valued Neural Networks](https://mlsanity.com/paper/2401.02277/) (2024)

## Built on by (most cited first, 4 of 4)

- [Polynomial Scaling is Possible For Neural Operator Approximations of Structured Families of BSDEs](https://mlsanity.com/paper/2410.14788/) (2024)
- [Topological DeepONets and a generalization of the Chen-Chen operator approximation theorem](https://mlsanity.com/paper/2603.11972/) (2026)
- [SVarM: Linear Support Varifold Machines for Classification and Regression on Geometric Data](https://mlsanity.com/paper/2506.01189/) (2025)
- [Universality of shallow and deep neural networks on non-Euclidean spaces](https://mlsanity.com/paper/2602.23381/) (2026)

## Abstract (arXiv)

We study feedforward neural networks with inputs from a topological vector space (TVS-FNNs). Unlike traditional feedforward neural networks, TVS-FNNs can process a broader range of inputs, including sequences, matrices, functions and more. We prove a universal approximation theorem for TVS-FNNs, which demonstrates their capacity to approximate any continuous function defined on this expanded input space.