Related papers: Approximation of Functionals by Neural Network wit…

Recently, deep learning has been widely applied in functional data analysis (FDA) with notable empirical success. However, the infinite dimensionality of functional data necessitates an effective dimension reduction approach for functional…

This paper proves an abstract theorem addressing in a unified manner two important problems in function approximation: avoiding curse of dimensionality and estimating the degree of approximation for out-of-sample extension in manifold…

The success of Neural networks in providing miraculous results when applied to a wide variety of tasks is astonishing. Insight in the working can be obtained by studying the universal approximation property of neural networks. It is proved…

We prove that for analytic functions in low dimension, the convergence rate of the deep neural network approximation is exponential.

Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks as the curse of dimensionality (CoD) cannot be evaded when trying to approximate even a single ReLU neuron…

In this paper we study shallow neural network functions which are linear combinations of compositions of activation and quadratic functions, replacing standard affine linear functions, often called neurons. We show the universality of this…

We investigate properties of neural networks that use both ReLU and $x^2$ as activation functions and build upon previous results to show that both analytic functions and functions in Sobolev spaces can be approximated by such networks of…

We use deep sparsely connected neural networks to measure the complexity of a function class in $L^2(\mathbb R^d)$ by restricting connectivity and memory requirement for storing the neural networks. We also introduce representation system -…

In this Master Thesis, we study the approximation capabilities of Neural Networks in the context of numerical resolution of elliptic PDEs and Approximation Theory. First of all, in Chapter 1, we introduce the mathematical definition of…

We study approximation and statistical learning properties of deep ReLU networks under structural assumptions that mitigate the curse of dimensionality. We prove minimax-optimal uniform approximation rates for $s$-H\"older smooth functions…

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…

Functions of one or more variables are usually approximated with a basis: a complete, linearly-independent system of functions that spans a suitable function space. The topic of this paper is the numerical approximation of functions using…

The approximation power of general feedforward neural networks with piecewise linear activation functions is investigated. First, lower bounds on the size of a network are established in terms of the approximation error and network depth…

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…

Quantum neural networks (QNNs) are an analog of classical neural networks in the world of quantum computing, which are represented by a unitary matrix with trainable parameters. Inspired by the universal approximation property of classical…

Deep neural networks (DNNs) achieve impressive results for complicated tasks like object detection on images and speech recognition. Motivated by this practical success, there is now a strong interest in showing good theoretical properties…

We develop a corrective mechanism for neural network approximation: the total available non-linear units are divided into multiple groups and the first group approximates the function under consideration, the second group approximates the…

Neural network width and depth are fundamental aspects of network topology. Universal approximation theorems provide that with increasing width or depth, there exists a neural network that approximates a function arbitrarily well. These…

We study the problem of approximating and estimating classification functions that have their decision boundary in the $RBV^2$ space. Functions of $RBV^2$ type arise naturally as solutions of regularized neural network learning problems and…

One of the key issues in the analysis of machine learning models is to identify the appropriate function space and norm for the model. This is the set of functions endowed with a quantity which can control the approximation and estimation…