Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural
  Network Derivatives
Yahong Yang Haizhao Yang Yang Xiang
This paper addresses the problem of nearly optimal Vapnik--Chervonenkis
dimension (VC-dimension) and pseudo-dimension estimations of the derivative
functions of deep neural networks (DNNs). Two important applications of these
estimations include: 1) Establishing a nearly tight approximation result of
DNNs in the Sobolev space; 2) Characterizing the generalization error of
machine learning methods with loss functions involving function derivatives.
This theoretical investigation fills the gap of learning error estimations for
a wide range of physics-informed machine learning models and applications
including generative models, solving partial differential equations, operator
learning, network compression, distillation, regularization, etc.