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Convolutional neural networks are the most widely used type of neural networks in applications. In mathematical analysis, however, mostly fully-connected networks are studied. In this paper, we establish a connection between both network…

In the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there…

Deep learning has exhibited remarkable results across diverse areas. To understand its success, substantial research has been directed towards its theoretical foundations. Nevertheless, the majority of these studies examine how well deep…

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…

Deep Reinforcement Learning (RL) powered by neural net approximation of the Q function has had enormous empirical success. While the theory of RL has traditionally focused on linear function approximation (or eluder dimension) approaches,…

Universal approximation theorem suggests that a shallow neural network can approximate any function. The input to neurons at each layer is a weighted sum of previous layer neurons and then an activation is applied. These activation…

Universal approximation theorems provide a mathematical explanation for the expressive power of neural networks. They assert that, under mild conditions on the activation function, feedforward neural networks are dense in broad function…

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…

In recent years there has been a considerable amount of research on local post hoc explanations for neural networks. However, work on building interpretable neural architectures has been relatively sparse. In this paper, we present a novel…

We study the approximation of functions which are invariant with respect to certain permutations of the input indices using flow maps of dynamical systems. Such invariant functions includes the much studied translation-invariant ones…

We analyze approximation rates of deep ReLU neural networks for Sobolev-regular functions with respect to weaker Sobolev norms. First, we construct, based on a calculus of ReLU networks, artificial neural networks with ReLU activation…

We investigate the concept of Best Approximation for Feedforward Neural Networks (FNN) and explore their convergence properties through the lens of Random Projection (RPNNs). RPNNs have predetermined and fixed, once and for all, internal…

In this paper it is shown that $C_\beta$-smooth functions can be approximated by deep neural networks with ReLU activation function and with parameters $\{0,\pm \frac{1}{2}, \pm 1, 2\}$. The $l_0$ and $l_1$ parameter norms of considered…

In this work, we examine the approximation capabilities of deep neural networks utilizing the Rectified Quadratic Unit (ReQU) activation function, defined as \(\max(0,x)^2\), for approximating H\"older-regular functions with respect to the…

This work addresses two fundamental limitations in neural network approximation theory. We demonstrate that a three-dimensional network architecture enables a significantly more efficient representation of sawtooth functions, which serves…

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…

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…

We consider approximations of general continuous functions on finite-dimensional cubes by general deep ReLU neural networks and study the approximation rates with respect to the modulus of continuity of the function and the total number of…

Neural networks hold great potential to act as approximate models of nonlinear dynamical systems, with the resulting neural approximations enabling verification and control of such systems. However, in safety-critical contexts, the use of…

The amount of information produced, whether by newspapers, blogs and social networks, or by monitoring systems, is increasing rapidly. Processing all this data in real-time, while taking into consideration advanced knowledge about the…