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It is well-known that neural networks are universal approximators, but that deeper networks tend in practice to be more powerful than shallower ones. We shed light on this by proving that the total number of neurons $m$ required to…

We describe generalizations of the universal approximation theorem for neural networks to maps invariant or equivariant with respect to linear representations of groups. Our goal is to establish network-like computational models that are…

Neural networks are one of the most popularly used methods in machine learning and artificial intelligence nowadays. Due to the universal approximation theorem (Hornik et al. (1989)), a neural network with one hidden layer can approximate…

Approximation theory is concerned with the ability to approximate functions by simpler and more easily calculated functions. The first question we ask in approximation theory concerns the {\it possibility of approximation}. Is the given…

This paper investigates the universal approximation capabilities of Hamiltonian Deep Neural Networks (HDNNs) that arise from the discretization of Hamiltonian Neural Ordinary Differential Equations. Recently, it has been shown that HDNNs…

We prove two universal approximation theorems for a range of dropout neural networks. These are feed-forward neural networks in which each edge is given a random $\{0,1\}$-valued filter, that have two modes of operation: in the first each…

We consider neural network approximation spaces that classify functions according to the rate at which they can be approximated (with error measured in $L^p$) by ReLU neural networks with an increasing number of coefficients, subject to…

This paper presents a novel framework of neural networks for isotropic hyperelasticity that enforces necessary physical and mathematical constraints while simultaneously satisfying the universal approximation theorem. The two key…

We give an abstract approach to approximations with a wide range of regularity classes $X$ in spaces of pseudocontinuable functions $K^p_\vartheta$, where $\vartheta$ is an inner function and $p>0$. More precisely, we demonstrate a general…

We study the approximation properties of neural ordinary differential equations (neural ODEs) in the space of continuous functions. Since a neural ODE requires input and output dimensions to be the same, while input and output dimensions of…

Deep neural networks have emerged as powerful tools for learning operators defined over infinite-dimensional function spaces. However, existing theories frequently encounter difficulties related to dimensionality and limited…

In this article, we analyze the approximation properties of the new family of Durrmeyer type exponential sampling operators. We derive the point-wise and uniform approximation theorem and Voronovskaya type theorem for these generalized…

We prove a characterization of the support of the law of the solution for a stochastic wave equation with two-dimensional space variable, driven by a noise white in time and correlated in space. The result is a consequence of an…

This paper discusses various theorems on the approximation capabilities of neural networks (NNs), which are known as universal approximation theorems (UATs). The paper gives a systematic overview of UATs starting from the preliminary…

The celebrated universal approximation theorems for neural networks roughly state that any reasonable function can be arbitrarily well-approximated by a network whose parameters are appropriately chosen real numbers. This paper examines the…

By universal formulas we understand parameterized analytic expressions that have a fixed complexity, but nevertheless can approximate any continuous function on a compact set. There exist various examples of such formulas, including some in…

In this paper, we explain the universal approximation capabilities of deep residual neural networks through geometric nonlinear control. Inspired by recent work establishing links between residual networks and control systems, we provide a…

We study the computational complexity of (deterministic or randomized) algorithms based on point samples for approximating or integrating functions that can be well approximated by neural networks. Such algorithms (most prominently…

In this short note, we give an elementary proof of a universal approximation theorem for neural networks with three hidden layers and increasing, continuous, bounded activation function. The result is weaker than the best known results, but…

In this work, we address the question whether a sufficiently deep quantum neural network can approximate a target function as accurate as possible. We start with simple but typical physical situations that the target functions are physical…