3 citations · 3 across the 11 of their papers we have counts for
12 papers
Convergence of Least Squares Methods for PDEs on R^d with Shallow ReLU^k Networks
Tong Mao, Lei Shi, Hanfei Zhou
We establish inverse inequalities in Gaussian Sobolev norms for shallow trial spaces. The estimates identify the sharp regularity threshold and quantify the depen…
Sharp Sobolev Approximation on General Domains by Linearized Shallow Networks with Analytic Activations
Jia Li, Tong Mao, Jinchao Xu
We study Sobolev approximation on bounded domains by linearized shallow neural networks whose inner parameters are prescribed independently of the target function. Our main step is…
Sharp Approximation on the Sphere by Linearized Shallow Networks with Analytic Activations
Jia Li, Tong Mao, Jinchao Xu
We study the spectral behavior and approximation properties of linearized shallow networks with analytic activation functions on the unit sphere . Motivated by earlier…
Optimal Neural Network Approximation via Empirical Least Squares with Deterministic Samples
Xinliang Liu, Tong Mao, Jinchao Xu
We develop a rigorous theory of discrete residual least-squares approximation for elliptic spectral equations using linearized ReLU neural networks on the sp…
Do Neural Networks Really Beat the Curse of Dimensionality? A Bit-Complexity View
Tong Mao, Jinchao Xu
Traditional approximation theory measures convergence rates in terms of the number of parameters or degrees of freedom. However, practical computation operates under finite precisi…
Solving High-Dimensional PDEs Using Linearized Neural Networks
Tong Mao, Jinchao Xu, Xiaofeng Xu
Linearized shallow neural networks that are constructed by fixing the hidden-layer parameters have recently shown strong performance in solving partial differential equations (PDEs…
Condition Numbers and Eigenvalue Spectra of Shallow Networks on Spheres
Xinliang Liu, Tong Mao, Jinchao Xu
We present an estimation of the condition numbers of the \emph{mass} and \emph{stiffness} matrices arising from shallow ReLU neural networks defined on the unit sphere~$\mathbb…
On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functions
Yulong Lu, Tong Mao, Jinchao Xu +1
Deep neural networks have been widely used as universal approximators for functions with inherent physical structures, including permutation symmetry. In this paper, we construct s…
Configuration-Dependent Lower Bounds for Approximation by Shallow ReLU Networks on the Sphere
Tong Mao, Jinchao Xu
We establish two related but logically distinct results for shallow ReLU neural networks on the unit sphere $\SS^d$. First, for an arbitrary set of inner neural-network paramet…
Integral Representations of Sobolev Spaces via ReLU Activation Function and Optimal Error Estimates for Linearized Networks
Xinliang Liu, Tong Mao, Jinchao Xu
This paper presents two main theoretical results concerning shallow neural networks with ReLU activation functions. We establish a novel integral representation for Sobolev spa…
Approximation Rates for Shallow ReLU Neural Networks on Sobolev Spaces via the Radon Transform
Tong Mao, Jonathan W. Siegel, Jinchao Xu
Let be a bounded domain. We consider the problem of how efficiently shallow neural networks with the ReLU activation function can approximate functions…
Expressivity and Approximation Properties of Deep Neural Networks with ReLU Activation
Juncai He, Tong Mao, Jinchao Xu
In this paper, we investigate the expressivity and approximation properties of deep neural networks employing the ReLU activation function for . Although deep ReLU ne…