Related papers: Approximation of Functionals by Neural Network wit…

The purpose of this article is to develop a machinery to study the capacity of deep neural networks (DNNs) to approximate high-dimensional functions. In particular, we show that DNNs have the expressive power to overcome the curse of…

In this paper we develop a new machinery to study the capacity of artificial neural networks (ANNs) to approximate high-dimensional functions without suffering from the curse of dimensionality. Specifically, we introduce a concept which we…

In this paper, we develop a framework for showing that neural networks can overcome the curse of dimensionality in different high-dimensional approximation problems. Our approach is based on the notion of a catalog network, which is a…

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…

We define a neural network in infinite dimensional spaces for which we can show the universal approximation property. Indeed, we derive approximation results for continuous functions from a Fr\'echet space $\X$ into a Banach space $\Y$. The…

This work explores the neural network approximation capabilities for functions within the spectral Barron space $\mathscr{B}^s$, where $s$ is the smoothness index. We demonstrate that for functions in $\mathscr{B}^{1/2}$, a shallow neural…

We study the approximation of measurable functions on the hypercube by functions arising from affine neural networks. Our main achievement is an approximation of any measurable function $f \colon W_n \to [-1,1]$ up to a prescribed precision…

In this article we identify a general class of high-dimensional continuous functions that can be approximated by deep neural networks (DNNs) with the rectified linear unit (ReLU) activation without the curse of dimensionality. In other…

Deep neural networks have been widely used as universal approximators for functions with inherent physical structures, including permutation symmetry. In this paper, we construct symmetric deep neural networks to approximate symmetric…

In this paper, we analyze the number of neurons and training parameters that a neural networks needs to approximate multivariate functions of bounded second mixed derivatives -- Korobov functions. We prove upper bounds on these quantities…

As demonstrated in many areas of real-life applications, neural networks have the capability of dealing with high dimensional data. In the fields of optimal control and dynamical systems, the same capability was studied and verified in many…

We propose a deep neural network architecture for storing approximate Lyapunov functions of systems of ordinary differential equations. Under a small-gain condition on the system, the number of neurons needed for an approximation of a…

Artificial neural networks (ANNs) have become a very powerful tool in the approximation of high-dimensional functions. Especially, deep ANNs, consisting of a large number of hidden layers, have been very successfully used in a series of…

Recently, it has been proposed in the literature to employ deep neural networks (DNNs) together with stochastic gradient descent methods to approximate solutions of PDEs. There are also a few results in the literature which prove that DNNs…

This paper establishes the nearly optimal rate of approximation for deep neural networks (DNNs) when applied to Korobov functions, effectively overcoming the curse of dimensionality. The approximation results presented in this paper are…

This paper quantitatively characterizes the approximation power of deep feed-forward neural networks (FNNs) in terms of the number of neurons. It is shown by construction that ReLU FNNs with width $\mathcal{O}\big(\max\{d\lfloor…

Is there any theoretical guarantee for the approximation ability of neural networks? The answer to this question is the "Universal Approximation Theorem for Neural Networks". This theorem states that a neural network is dense in a certain…

Motivated by the abundance of functional data such as time series and images, there has been a growing interest in integrating such data into neural networks and learning maps from function spaces to R (i.e., functionals). In this paper, we…

This paper investigates the approximation properties of shallow neural networks with activation functions that are powers of exponential functions. It focuses on the dependence of the approximation rate on the dimension and the smoothness…

Universal approximation theorems are the foundations of classical neural networks, providing theoretical guarantees that the latter are able to approximate maps of interest. Recent results have shown that this can also be achieved in a…