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: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
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
arXiv:2609.15355v1 Announce Type: cross
Abstract: We study the uniform approximation of smooth scalar-valued functionals on an infinite-dimensional separable Hilbert space by deep ReLU neural network...
By Shuhao Jiao
arXiv:2406. 12264v5 Announce Type: replace-cross Abstract: We obtain a new universal approximation theorem for continuous (possibly nonlinear) operators on arbitrary Banach spaces using the Leray-Schauder mapping.
By Emanuele Zappala
arXiv:2609.15355v2 Announce Type: replace-cross
Abstract: We study the uniform approximation of smooth scalar-valued functionals on an infinite-dimensional separable Hilbert space by ReLU neural netw...
By Shuhao Jiao