Related papers: Differentiable Neural Networks with RePU Activatio…

Deep neural network with rectified linear units (ReLU) is getting more and more popular recently. However, the derivatives of the function represented by a ReLU network are not continuous, which limit the usage of ReLU network to situations…

The Rectified Power Unit (RePU) activation function, a differentiable generalization of the Rectified Linear Unit (ReLU), has shown promise in constructing neural networks due to its smoothness properties. However, deep RePU networks often…

Deep neural networks with rectified linear units (ReLU) are getting more and more popular due to their universal representation power and successful applications. Some theoretical progress regarding the approximation power of deep ReLU…

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…

In this paper we investigate the family of functions representable by deep neural networks (DNN) with rectified linear units (ReLU). We give an algorithm to train a ReLU DNN with one hidden layer to *global optimality* with runtime…

In this effort, we derive a formula for the integral representation of a shallow neural network with the Rectified Power Unit activation function. Mainly, our first result deals with the univariate case of representation capability of RePU…

We consider approximation rates of sparsely connected deep rectified linear unit (ReLU) and rectified power unit (RePU) neural networks for functions in Besov spaces $B^\alpha_{q}(L^p)$ in arbitrary dimension $d$, on general domains. We…

Implementing deep neural networks for learning the solution maps of parametric partial differential equations (PDEs) turns out to be more efficient than using many conventional numerical methods. However, limited theoretical analyses have…

Recently, several deep learning (DL) methods for approximating high-dimensional partial differential equations (PDEs) have been proposed. The interest that these methods have generated in the literature is in large part due to simulations…

Deep neural networks (DNNs) with ReLU activation function are proved to be able to express viscosity solutions of linear partial integrodifferental equations (PIDEs) on state spaces of possibly high dimension $d$. Admissible PIDEs comprise…

In this article we study high-dimensional approximation capacities of shallow and deep artificial neural networks (ANNs) with the rectified linear unit (ReLU) activation. In particular, it is a key contribution of this work to reveal that…

We can compare the expressiveness of neural networks that use rectified linear units (ReLUs) by the number of linear regions, which reflect the number of pieces of the piecewise linear functions modeled by such networks. However,…

Activation function is crucial to the recent successes of deep neural networks. In this paper, we first propose a new activation function, Multiple Parametric Exponential Linear Units (MPELU), aiming to generalize and unify the rectified…

In this paper we consider PIDEs with gradient-independent Lipschitz continuous nonlinearities and prove that deep neural networks with ReLU activation function can approximate solutions of such semilinear PIDEs without curse of…

Motivated by the growing theoretical understanding of neural networks that employ the Rectified Linear Unit (ReLU) as their activation function, we revisit the use of ReLU activation functions for learning implicit neural representations…

In this paper, we focus on fully connected deep neural networks utilizing the Rectified Linear Unit (ReLU) activation function for nonparametric estimation. We derive non-asymptotic bounds that lead to convergence rates, addressing both…

In recent years, functional neural networks have been proposed and studied in order to approximate nonlinear continuous functionals defined on $L^p([-1, 1]^s)$ for integers $s\ge1$ and $1\le p<\infty$. However, their theoretical properties…

In this paper, we investigate the relationship between deep neural networks (DNN) with rectified linear unit (ReLU) function as the activation function and continuous piecewise linear (CPWL) functions, especially CPWL functions from the…

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…

Activation functions have been shown to affect the performance of deep neural networks significantly. While the Rectified Linear Unit (ReLU) remains the dominant choice in practice, the optimal activation function for deep neural networks…