155 citations · 987 across the 104 of their papers we have counts for
12 papers · 1 filter
An Interpretive Constrained Linear Model for ResNet and MgNet
Juncai He, Jinchao Xu, Lian Zhang +1
We propose a constrained linear data-feature-mapping model as an interpretable mathematical model for image classification using a convolutional neural network (CNN). From this vie…
Approximation Properties of Deep ReLU CNNs
Juncai He, Lin Li, Jinchao Xu
This paper focuses on establishing approximation properties for deep ReLU convolutional neural networks (CNNs) in two-dimensional space. The analysis is based on a decomposit…
Integral boundary conditions in phase field models
Xiaofeng Xu, Lian Zhang, Yin Shi +2
Modeling the chemical, electric, and thermal transport as well as phase transitions and the accompanying mesoscale microstructure evolution within a material in an electronic devic…
Greedy Training Algorithms for Neural Networks and Applications to PDEs
Jonathan W. Siegel, Qingguo Hong, Xianlin Jin +2
Recently, neural networks have been widely applied for solving partial differential equations (PDEs). Although such methods have been proven remarkably successful on practical engi…
Characterization of the Variation Spaces Corresponding to Shallow Neural Networks
Jonathan W. Siegel, Jinchao Xu
We study the variation space corresponding to a dictionary of functions in for a bounded domain . Specifically, we compare the variation space, whic…
Optimal Convergence Rates for the Orthogonal Greedy Algorithm
Jonathan W. Siegel, Jinchao Xu
We analyze the orthogonal greedy algorithm when applied to dictionaries whose convex hull has small entropy. We show that if the metric entropy of the convex hull of $…
Sharp Lower Bounds on the Approximation Rate of Shallow Neural Networks
Jonathan W. Siegel, Jinchao Xu
We consider the approximation rates of shallow neural networks with respect to the variation norm. Upper bounds on these rates have been established for sigmoidal and ReLU activati…
ReLU Deep Neural Networks from the Hierarchical Basis Perspective
Juncai He, Lin Li, Jinchao Xu
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 sch…
A Priori Analysis of Stable Neural Network Solutions to Numerical PDEs
Qingguo Hong, Jonathan W. Siegel, Jinchao Xu
Methods for solving PDEs using neural networks have recently become a very important topic. We provide an a priori error analysis for such methods which is based on the $\mathcal{K…
Robust BPX Preconditioner for Fractional Laplacians on Bounded Lipschitz Domains
Juan Pablo Borthagaray, Ricardo H. Nochetto, Shuonan Wu +1
We propose and analyze a robust BPX preconditioner for the integral fractional Laplacian on bounded Lipschitz domains. For either quasi-uniform grids or graded bisection grids, we…
An Extended Galerkin analysis in finite element exterior calculus
Qingguo Hong, Yuwen Li, Jinchao Xu
For the Hodge--Laplace equation in finite element exterior calculus, we introduce several families of discontinuous Galerkin methods in the extended Galerkin framework. For contrac…
Sharp Bounds on the Approximation Rates, Metric Entropy, and -widths of Shallow Neural Networks
Jonathan W. Siegel, Jinchao Xu
In this article, we study approximation properties of the variation spaces corresponding to shallow neural networks with a variety of activation functions. We introduce two main to…