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