# Weighted universal approximation of differentiable maps on infinite-dimensional manifolds

arXiv: https://arxiv.org/abs/2606.09820 · HTML: https://mlsanity.com/paper/2606.09820/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 124 references that are in the map.
- First posted on arXiv 8 Jun 2026.

## Builds on (most cited first, 4 of 4)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [An Approximation Theory for Metric Space-Valued Functions With A View Towards Deep Learning](https://mlsanity.com/paper/2304.12231/) (2023)
- [Generative Neural Operators of Log-Complexity Can Simultaneously Solve Infinitely Many Convex Programs](https://mlsanity.com/paper/2508.14995/) (2025)
- [Global universal approximation with Brownian signatures](https://mlsanity.com/paper/2512.16396/) (2025)

## Built on by (most cited first, 2 of 2)

- [NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes](https://mlsanity.com/paper/2607.14361/) (2026)
- [Neural operators approximate strongly continuous convex monotone semigroups](https://mlsanity.com/paper/2609.02727/) (2026)

## Abstract (arXiv)

We generalize the universal approximation theorem for functional input neural networks (FNN) to differentiable maps by including the approximation of the derivatives. A FNN maps the input from a possibly infinite-dimensional weighted manifold to the real-valued hidden layer, on which a non-linear scalar activation function is applied, and then returns the output into a Banach space via some linear readouts. By proving a weighted Nachbin theorem, we establish a universal approximation theorem for differentiable maps, which goes beyond the usual formulation on compact sets and also includes the approximation of the derivatives. This leads us to approximation results for non-anticipative functionals including the horizontal and vertical derivatives. As a further application, we show that linear functions of the signature are able to approximate path space functionals including their directional derivatives.