Related papers: Approximation bounds for convolutional neural netw…
Deep neural networks are powerful learning models that achieve state-of-the-art performance on many computer vision, speech, and language processing tasks. In this paper, we study a fundamental question that arises when designing deep…
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…
A burgeoning line of research leverages deep neural networks to approximate the solutions to high dimensional PDEs, opening lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of…
When optimizing convolutional neural networks (CNN) for a specific image-based task, specialists commonly overshoot the number of convolutional layers in their designs. By implication, these CNNs are unnecessarily resource intensive to…
Operator convex functions defined on the positive half-line play a prominent role in the theory of quantum information, where they are used to define quantum $f$-divergences. Such functions admit integral representations in terms of…
The use of neural networks to solve differential equations, as an alternative to traditional numerical solvers, has increased recently. However, error bounds for the obtained solutions have only been developed for certain equations. In this…
Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…
Neural networks are known to be effective function approximators. Recently, deep neural networks have proven to be very effective in pattern recognition, classification tasks and human-level control to model highly nonlinear realworld…
The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are…
Convolutional Neural Networks have revolutionized vision applications. There are image domains and representations, however, that cannot be handled by standard CNNs (e.g., spherical images, superpixels). Such data are usually processed…
This paper aims to accelerate the test-time computation of convolutional neural networks (CNNs), especially very deep CNNs that have substantially impacted the computer vision community. Unlike previous methods that are designed for…
We present ApproxConv, a novel method for compressing the layers of a convolutional neural network. Reframing conventional discrete convolution as continuous convolution of parametrised functions over space, we use functional approximations…
We show the existence of a deep neural network capable of approximating a wide class of high-dimensional approximations. The construction of the proposed neural network is based on a quasi-optimal polynomial approximation. We show that this…
We propose a novel class of neural network-like parametrized functions, i.e., general transformation neural networks (GTNNs), for high-dimensional approximation. Conventional deep neural networks sometimes perform less accurately on…
We describe an approach for estimating the statistical risk of deep networks that contain a mix of learned and known operators. Building on the maximal training error bounds previously established for known operator learning, we derive a…
In this paper, we consider approximating the parameter-to-solution maps of parametric partial differential equations (PPDEs) using deep neural networks (DNNs). We propose an efficient approach combining reduced collocation methods (RCMs)…
This article presents a reduced-order modeling methodology via deep convolutional neural networks (CNNs) for shape optimization applications. The CNN provides a nonlinear mapping between the shapes and their associated attributes while…
We explore the approximation capabilities of Transformer networks for H\"older and Sobolev functions, and apply these results to address nonparametric regression estimation with dependent observations. First, we establish novel upper bounds…
Within the world of machine learning there exists a wide range of different methods with respective advantages and applications. This paper seeks to present and discuss one such method, namely Convolutional Neural Networks (CNNs). CNNs are…
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…