30 lines
2.0 KiB
Markdown
30 lines
2.0 KiB
Markdown
---
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title: "Weighted Universal Approximation of Differentiable Maps on Infinite-Dimensional Manifolds"
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source_url: https://arxiv.org/abs/2606.09820
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ingested: 2026-06-17
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sha256: <computed>
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---
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# Weighted Universal Approximation of Differentiable Maps on Infinite-Dimensional Manifolds
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**Authors:** Philipp Schmocker, Josef Teichmann
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**arXiv:** 2606.09820v1 [math.FA] (2026-06-08)
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**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.
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## Abstract
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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.
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## Key Concepts
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- [[functional-input-neural-networks|函数输入神经网络 (FNN)]]
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- [[universal-approximation-theorem|通用逼近定理 (UAT)]]
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- [[nachbin-theorem|Nachbin 定理]] / [[weighted-spaces|加权空间]]
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- [[infinite-dimensional-manifolds|无限维流形]]
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- [[bastiani-calculus|Bastiani 微积分]]
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- [[non-anticipative-functionals|非预期泛函]]
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- [[signature|签名 (Signature)]]
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- [[rough-path-theory|粗糙路径理论]]
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