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In this paper, we develop a theory about the relationship between $G$-invariant/equivariant functions and deep neural networks for finite group $G$. Especially, for a given $G$-invariant/equivariant function, we construct its universal…

We introduce a mild generative variant of the classical neural operator model, which leverages Kolmogorov--Arnold networks to solve infinite families of second-order backward stochastic differential equations ($2$BSDEs) on regular bounded…

In this paper we study the theory of the so-called Kantorovich max-product neural network operators in the setting of Orlicz spaces $L^{\varphi}$. The results here proved, extend those given by Costarelli and Vinti in Result Math., 2016, to…

Artificial neural network operators (ANNOs) have been widely used for approximating deterministic input-output functions; however, their extension to random dynamics remains comparatively unexplored. In this paper, we construct a new class…

We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also find the order of this approximation by using…

We establish in this work approximation results of deep neural networks for smooth functions measured in Sobolev norms, motivated by recent development of numerical solvers for partial differential equations using deep neural networks. {Our…

We identify various classes of neural networks that are able to approximate continuous functions locally uniformly subject to fixed global linear growth constraints. For such neural networks the associated neural stochastic differential…

Multiplication layers are a key component in various influential neural network modules, including self-attention and hypernetwork layers. In this paper, we investigate the approximation capabilities of deep neural networks with…

In this paper, we establish universal approximation theorems for neural networks applied to general nonlinear ill-posed operator equations. In addition to the approximation error, the measurement error is also taken into account in our…

Overparameterized neural networks enjoy great representation power on complex data, and more importantly yield sufficiently smooth output, which is crucial to their generalization and robustness. Most existing function approximation…

Superconformal transformations are derived for the $\N=2,4 supermultiplets corresponding to the simplest chiral primary operators. These are applied to two, three and four point correlation functions. When $\N=4$, results are obtained for…

There has been a growing interest in expressivity of deep neural networks. However, most of the existing work about this topic focuses only on the specific activation function such as ReLU or sigmoid. In this paper, we investigate the…

In this paper, we investigate the approximation properties of the summation-integral type operators as defined by Mishra et al. (Boll. Unione Mat. Ital. (2016) 8:297-305) and determine the local results as well as prove the convergence…

In this work, we consider the approximation of a large class of bounded functions, with minimal regularity assumptions, by ReLU neural networks. We show that the approximation error can be bounded from above by a quantity proportional to…

Proximal operators are ubiquitous in inverse problems, commonly appearing as part of algorithmic strategies to regularize problems that are otherwise ill-posed. Modern deep learning models have been brought to bear for these tasks too, as…

Neural operators improve conventional neural networks by expanding their capabilities of functional mappings between different function spaces to solve partial differential equations (PDEs). One of the most notable methods is the Fourier…

We propose a novel class of neural network-like parametrized functions, i.e., general transformation neural networks (GTNNs), for high-dimensional approximation. Conventional deep neural networks sometimes perform less accurately on…

Fourier neural operators (FNOs) have recently been proposed as an effective framework for learning operators that map between infinite-dimensional spaces. We prove that FNOs are universal, in the sense that they can approximate any…

Families of quasi-permutable normal operators in octonion Hilbert spaces are investigated. Their spectra are studied. Multiparameter semigroups of such operators are considered. A non-associative analog of Stone's theorem is proved.

The problem of reconstructing functions from their asymptotic expansions in powers of a small variable is addressed by deriving a novel type of approximants. The derivation is based on the self-similar approximation theory, which presents…