Related papers: Deep neural network approximation theory for high-…
In this paper, we establish that for a wide class of controlled stochastic differential equations (SDEs) with stiff coefficients, the value functions of corresponding zero-sum games can be represented by a deep artificial neural network…
A key challenge in scientific machine learning is solving partial differential equations (PDEs) on complex domains, where the curved geometry complicates the approximation of functions and their derivatives required by differential…
In an ever expanding set of research and application areas, deep neural networks (DNNs) set the bar for algorithm performance. However, depending upon additional constraints such as processing power and execution time limits, or…
One of the most prominent attributes of Neural Networks (NNs) constitutes their capability of learning to extract robust and descriptive features from high dimensional data, like images. Hence, such an ability renders their exploitation as…
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…
We study ReLU deep neural networks (DNNs) by investigating their connections with the hierarchical basis method in finite element methods. First, we show that the approximation schemes of ReLU DNNs for $x^2$ and $xy$ are composition…
In studying the expressiveness of neural networks, an important question is whether there are functions which can only be approximated by sufficiently deep networks, assuming their size is bounded. However, for constant depths, existing…
In this paper we present a class of convolutional neural networks (CNNs) called non-overlapping CNNs in the study of approximation capabilities of CNNs. We prove that such networks with sigmoidal activation function are capable of…
Deep neural networks (DNNs) generate much richer function spaces than shallow networks. Since the function spaces induced by shallow networks have several approximation theoretic drawbacks, this explains, however, not necessarily the…
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…
In this paper we introduce new bounds on the approximation of functions in deep networks and in doing so introduce some new deep network architectures for function approximation. These results give some theoretical insight into the success…
This paper develops a randomized approach for incrementally building deep neural networks, where a supervisory mechanism is proposed to constrain the random assignment of the weights and biases, and all the hidden layers have direct links…
The developments of deep neural networks (DNN) in recent years have ushered a brand new era of artificial intelligence. DNNs are proved to be excellent in solving very complex problems, e.g., visual recognition and text understanding, to…
In this paper we consider Deep Neural Networks (DNNs) with a smooth activation function as surrogates for high-dimensional functions that are somewhat smooth but costly to evaluate. We consider the standard (non-periodic) DNNs as well as…
Deep neural networks have emerged as powerful tools for learning operators defined over infinite-dimensional function spaces. However, existing theories frequently encounter difficulties related to dimensionality and limited…
Deep Neural Networks (DNNs) excel at many tasks, often rivaling or surpassing human performance. Yet their internal processes remain elusive, frequently described as "black boxes." While performance can be refined experimentally, achieving…
This paper concerns the universal approximation property with neural networks in variable Lebesgue spaces. We show that, whenever the exponent function of the space is bounded, every function can be approximated with shallow neural networks…
Deep learning based on deep neural networks of various structures and architectures has been powerful in many practical applications, but it lacks enough theoretical verifications. In this paper, we consider a family of deep convolutional…
Deep neural networks (DNNs) have garnered significant attention in various fields of science and technology in recent years. Activation functions define how neurons in DNNs process incoming signals for them. They are essential for learning…
Artificial intelligence, particularly the subfield of machine learning, has seen a paradigm shift towards data-driven models that learn from and adapt to data. This has resulted in unprecedented advancements in various domains such as…