相关论文: Error bounds for approximations with deep ReLU net…

We develop a quantitative approximation theory for shallow neural networks using tools from time-frequency analysis. Working in weighted modulation spaces $M^{p,q}_m(\mathbf{R}^{d})$, we prove dimension-independent approximation rates in…

A Random Vector Functional Link (RVFL) network is a depth-2 neural network with random inner weights and biases. Only the outer weights of such an architecture are to be learned, so the learning process boils down to a linear optimization…

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 study the sample complexity of learning one-hidden-layer convolutional neural networks (CNNs) with non-overlapping filters. We propose a novel algorithm called approximate gradient descent for training CNNs, and show that, with high…

Recent theoretical work has demonstrated that deep neural networks have superior performance over shallow networks, but their training is more difficult, e.g., they suffer from the vanishing gradient problem. This problem can be typically…

Single hidden layer feedforward neural networks can represent multivariate functions that are sums of ridge functions. These ridge functions are defined via an activation function and customizable weights. The paper deals with best…

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…

This work focuses on the analysis of fully connected feed forward ReLU neural networks as they approximate a given, smooth function. In contrast to conventionally studied universal approximation properties under increasing architectures,…

Transformer networks have achieved remarkable empirical success across a wide range of applications, yet their theoretical expressive power remains insufficiently understood. In this paper, we study the expressive capabilities of…

\citet{farrell2021deep} establish non-asymptotic high-probability bounds for general deep feedforward neural network (with rectified linear unit activation function) estimators, with \citet[Theorem 1]{farrell2021deep} achieving a suboptimal…

Recent years have witnessed a resurgence in using ReLU neural networks (NNs) to represent model predictive control (MPC) policies. However, determining the required network complexity to ensure closed-loop performance remains a fundamental…

We investigate the implications of removing bias in ReLU networks regarding their expressivity and learning dynamics. We first show that two-layer bias-free ReLU networks have limited expressivity: the only odd function two-layer bias-free…

In a previous study [B. Li, S. Tang and H. Yu, Commun. Comput. Phy. 27(2):379-411, 2020], it is shown that deep neural networks built with rectified power units (RePU) as activation functions can give better approximation for sufficient…

For several classes of neural PDE solvers (Deep Ritz, PINNs, DeepONets), the ability to approximate the solution or solution operator to a partial differential equation (PDE) hinges on the abilitiy of a neural network to approximate the…

Deep learning training training algorithms are a huge success in recent years in many fields including speech, text,image video etc. Deeper and deeper layers are proposed with huge success with resnet structures having around 152 layers.…

We prove that the set of functions representable by ReLU neural networks with integer weights strictly increases with the network depth while allowing arbitrary width. More precisely, we show that $\lceil\log_2(n)\rceil$ hidden layers are…

This paper investigates the relationship between the universal approximation property of deep neural networks and topological characteristics of datasets. Our primary contribution is to introduce data topology-dependent upper bounds on the…

We study the power of deep neural networks (DNNs) with sigmoid activation function. Recently, it was shown that DNNs approximate any $d$-dimensional, smooth function on a compact set with a rate of order $W^{-p/d}$, where $W$ is the number…

We study the approximation capacity of deep ReLU recurrent neural networks (RNNs) and explore the convergence properties of nonparametric least squares regression using RNNs. We derive upper bounds on the approximation error of RNNs for…

This paper develops simple feed-forward neural networks that achieve the universal approximation property for all continuous functions with a fixed finite number of neurons. These neural networks are simple because they are designed with a…