Sobolev Spaces

Approximation and Generalization Errors in Deep Neural Networks for Sobolev Spaces measured by Sobolev Norms

Dr. Yahong Yang, Department of Mathematics, Penn State University

Feb 14, 16:00 - 17:00

KAUST

deep neural networks approximation Sobolev Spaces

Abstract In this presentation, we initially discuss the approximation capabilities of deep neural networks (DNNs) with ReLU and the square of ReLU as activation functions for Sobolev functions measured by the Sobolev norms \(W^{m,p}\) where \(m \ge 1\). Subsequently, we consider how to address the issue of the curse of dimensionality for DNNs’ approximation. Finally, we analyze the generalization errors associated with DNNs using such Sobolev loss functions. Additionally, we provide recommendations on when to opt for deeper NNs versus wider NNs, considering factors such as the number of sample