Related papers: Neural Networks in Fr\'echet spaces

The capability of recurrent neural networks to approximate trajectories of a random dynamical system, with random inputs, on non-compact domains, and over an indefinite or infinite time horizon is considered. The main result states that…

To understand the training dynamics of neural networks (NNs), prior studies have considered the infinite-width mean-field (MF) limit of two-layer NN, establishing theoretical guarantees of its convergence under gradient flow training as…

Molecular modeling is an important topic in drug discovery. Decades of research have led to the development of high quality scalable molecular force fields. In this paper, we show that neural networks can be used to train a universal…

Fully connected deep neural networks are successfully applied to classification and function approximation problems. By minimizing the cost function, i.e., finding the proper weights and biases, models can be built for accurate predictions.…

Neural networks have gained much interest because of their effectiveness in many applications. However, their mathematical properties are generally not well understood. If there is some underlying geometric structure inherent to the data or…

In the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there…

Complex-valued neural networks have attracted increasing attention in recent years, while it remains open on the advantages of complex-valued neural networks in comparison with real-valued networks. This work takes one step on this…

The classical development of neural networks has been primarily for mappings between a finite-dimensional Euclidean space and a set of classes, or between two finite-dimensional Euclidean spaces. The purpose of this work is to generalize…

To characterize the function space explored by neural networks (NNs) is an important aspect of learning theory. In this work, noticing that a multi-layer NN generates implicitly a hierarchy of reproducing kernel Hilbert spaces (RKHSs) -…

In recent years there has been a considerable amount of research on local post hoc explanations for neural networks. However, work on building interpretable neural architectures has been relatively sparse. In this paper, we present a novel…

We aimed to explore the capability of deep learning to approximate the function instantiated by biological neural circuits-the functional connectome. Using deep neural networks, we performed supervised learning with firing rate observations…

In this study, we explore the integration of Neural Networks, a powerful class of functions known for their exceptional approximation capabilities. Our primary emphasis is on the integration of multi-layer Neural Networks, a challenging…

The problem of extending a function $f$ defined on a training data $\mathcal{C}$ on an unknown manifold $\mathbb{X}$ to the entire manifold and a tubular neighborhood of this manifold is considered in this paper. For $\mathbb{X}$ embedded…

Neural ODEs and i-ResNet are recently proposed methods for enforcing invertibility of residual neural models. Having a generic technique for constructing invertible models can open new avenues for advances in learning systems, but so far…

Motivated by the developing mathematics of deep learning, we build universal functions approximators of continuous maps between arbitrary Polish metric spaces $\mathcal{X}$ and $\mathcal{Y}$ using elementary functions between Euclidean…

This paper extends the universal approximation property of single-hidden-layer feedforward neural networks beyond compact domains, which is of particular interest for the approximation within weighted $C^k$-spaces and weighted Sobolev…

A novel artificial neural network method is proposed for solving Cauchy inverse problems. It allows multiple hidden layers with arbitrary width and depth, which theoretically yields better approximations to the inverse problems. In this…

Operator learning refers to the application of ideas from machine learning to approximate (typically nonlinear) operators mapping between Banach spaces of functions. Such operators often arise from physical models expressed in terms of…

We prove two universal approximation theorems for a range of dropout neural networks. These are feed-forward neural networks in which each edge is given a random $\{0,1\}$-valued filter, that have two modes of operation: in the first each…

Group symmetry is inherent in a wide variety of data distributions. Data processing that preserves symmetry is described as an equivariant map and often effective in achieving high performance. Convolutional neural networks (CNNs) have been…