相关论文: Approximation Capabilities of Neural ODEs and Inve…

Invertible neural networks (INNs) are neural network architectures with invertibility by design. Thanks to their invertibility and the tractability of Jacobian, INNs have various machine learning applications such as probabilistic modeling,…

Neural ordinary differential equations (NODEs) is an invertible neural network architecture promising for its free-form Jacobian and the availability of a tractable Jacobian determinant estimator. Recently, the representation power of NODEs…

In this article, we explore the effects of memory terms in continuous-layer Deep Residual Networks by studying Neural ODEs (NODEs). We investigate two types of models. On one side, we consider the case of Residual Neural Networks with…

A class of neural networks that gained particular interest in the last years are neural ordinary differential equations (neural ODEs). We study input-output relations of neural ODEs using dynamical systems theory and prove several results…

Measure-preserving neural networks are well-developed invertible models, however, their approximation capabilities remain unexplored. This paper rigorously analyses the approximation capabilities of existing measure-preserving neural…

In this paper, we explain the universal approximation capabilities of deep residual neural networks through geometric nonlinear control. Inspired by recent work establishing links between residual networks and control systems, we provide a…

We introduce a class of fully-connected neural networks whose activation functions, rather than being pointwise, rescale feature vectors by a function depending only on their norm. We call such networks radial neural networks, extending…

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…

We demonstrate that a very deep ResNet with stacked modules with one neuron per hidden layer and ReLU activation functions can uniformly approximate any Lebesgue integrable function in $d$ dimensions, i.e. $\ell_1(\mathbb{R}^d)$. Because of…

In numerous robotics and mechanical engineering applications, among others, data is often constrained on smooth manifolds due to the presence of rotational degrees of freedom. Common datadriven and learning-based methods such as neural…

In this paper we consider the limiting case of neural networks (NNs) architectures when the number of neurons in each hidden layer and the number of hidden layers tend to infinity thus forming a continuum, and we derive approximation errors…

We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with…

We study shallow and deep neural networks whose inputs range over a general topological space. The model is built from a prescribed family of continuous feature maps and reduces to multilayer feedforward networks in the Euclidean case. We…

Invertible neural networks (INNs) represent an important class of deep neural network architectures that have been widely used in several applications. The universal approximation properties of INNs have also been established recently.…

The study of universal approximation of arbitrary functions $f: \mathcal{X} \to \mathcal{Y}$ by neural networks has a rich and thorough history dating back to Kolmogorov (1957). In the case of learning finite dimensional maps, many authors…

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…

Reaction-diffusion systems represent one of the most fundamental formulations used to describe a wide range of physical, chemical, and biological processes. With the increasing adoption of neural networks, recent research has focused on…

Residual networks (ResNets) are a deep learning architecture that substantially improved the state of the art performance in certain supervised learning tasks. Since then, they have received continuously growing attention. ResNets have a…

We introduce a technique that enables Neural-ODEs to approximate arbitrary velocity fields with a priori planted fixed-points. Specifically, a recipe is given to explicitly accommodate for a finite collection of points in the reference…

The universal approximation theorem states that a neural network with one hidden layer can approximate continuous functions on compact sets with any desired precision. This theorem supports using neural networks for various applications,…