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.
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.
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.
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.
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...
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.
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...
arXiv:2606. 20325v1 Announce Type: new Abstract: Classical approximation theorems ask for a new neural network whenever the target accuracy is improved.
arXiv:2605. 01702v2 Announce Type: replace Abstract: Theoretical studies show that for any differentiable function on a compact domain, there exists a neural network that approximates both the function values and gradients.
The Universal Approximation Theorem states that a neural network with a single hidden layer is sufficient to approximate any continuous univariate function on a compact domain to arbitrary error. However, the uniqueness of such neural network representations is not guaranteed, raising questions about practical identifiability.
CAGE-NAS introduces a method for deciding when to grow a neural network by evaluating an admissibility criterion on approximations of the functional gradient. If the current architecture allows a certified Functional Gradient Descent step, it remains unchanged; otherwise, a function-preserving expansion is performed and re-evaluated. Using a tangent-space-based instance with regularized projection, the approach achieves architectures that rank above the 99.8th percentile in held‑out RMSE among all admissible alternatives within the same parameter budget, without enumerating them during growth.
arXiv:2403.12187v2 Announce Type: replace-cross Abstract: Motivated by the abundance of functional data, such as time series and images, we study the approximation and statistical learning of nonline...
arXiv:2510. 15814v2 Announce Type: replace-cross Abstract: Universality results for equivariant neural networks remain rare.