arXiv Machine Learning By Philipp Schmocker, Josef Teichmann

Weighted universal approximation of differentiable maps on infinite-dimensional manifolds

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arXiv:2606. 09820v1 Announce Type: cross Abstract: We generalize the universal approximation theorem for functional input neural networks (FNN) to differentiable maps by including the approximation of the derivatives.

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arXiv Machine Learning
Jul 16

New universal operator approximation theorem for encoder-decoder architectures

arXiv:2503. 24092v2 Announce Type: replace-cross Abstract: Motivated by the rapidly growing field of mathematics for operator approximation with neural networks, we present a novel universal operator approximation theorem for broad classes of encoder-decoder architectures and a wide range of input and output spaces.

By Janek G\"odeke, Pascal Fernsel
arXiv AI
Jul 9

Understanding Two-Layer Neural Networks with Smooth Activation Functions

arXiv:2507. 14177v2 Announce Type: replace-cross Abstract: This paper aims to understand the training solution, which is obtained by the back-propagation algorithm, of two-layer neural networks whose hidden layer is composed of the units with smooth activation functions, including the usual sigmoid type most commonly used before the advent of ReLUs.

By Changcun Huang
arXiv AI
Sep 2

Universal Approximation of Nonlinear Operators and Their Derivatives

The paper establishes the first Universal Approximation Theorems for k‑times differentiable nonlinear operators and their derivatives in general Banach spaces, extending classical results to infinite‑dimensional settings. It introduces Derivative‑Informed Operator Learning (DIOL) and formulates Bastiani–Sobolev training for this framework, covering architectures such as DeepONets, Deep‑H‑ONets, and PCA‑Nets. The work also outlines applications to high‑order accuracy in operator learning, constrained optimization in Banach spaces, and numerical methods for infinite‑dimensional PDEs.

By Filippo de Feo