60 citations · 202 across the 22 of their papers we have counts for
6 papers · 1 filter
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…
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…
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…