9 papers
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…
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…
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…
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…