---
title: "Weighted Universal Approximation of Differentiable Maps on Infinite-Dimensional Manifolds"
source_url: https://arxiv.org/abs/2606.09820
ingested: 2026-06-17
sha256: <computed>
---

# Weighted Universal Approximation of Differentiable Maps on Infinite-Dimensional Manifolds

**Authors:** Philipp Schmocker, Josef Teichmann

**arXiv:** 2606.09820v1 [math.FA] (2026-06-08)

**Keywords:** Machine learning, neural operator, Universal approximation, weighted approximation, infinite-dimensional manifold, locally convex topological vector space, bounded approximation property, Stone-Weierstrass theorem, Nachbin theorem, Tauberian theorem, non-anticipative functional, rough path, signature.

## Abstract

Generalizes 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, establishes a universal approximation theorem (UAT) for differentiable maps, which goes beyond the usual formulation on compact sets and also includes the approximation of the derivatives. Leads to approximation results for non-anticipative functionals including the horizontal and vertical derivatives. Also shows that linear functions of the signature are able to approximate path space functionals including their directional derivatives.

## Key Concepts

- [[functional-input-neural-networks|函数输入神经网络 (FNN)]]
- [[universal-approximation-theorem|通用逼近定理 (UAT)]]
- [[nachbin-theorem|Nachbin 定理]] / [[weighted-spaces|加权空间]]
- [[infinite-dimensional-manifolds|无限维流形]]
- [[bastiani-calculus|Bastiani 微积分]]
- [[non-anticipative-functionals|非预期泛函]]
- [[signature|签名 (Signature)]]
- [[rough-path-theory|粗糙路径理论]]