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This paper addresses the problem of nearly optimal Vapnik--Chervonenkisdimension (VC-dimension) and pseudo-dimension estimations of the derivativefunctions of deep neural networks (DNNs). Two important applications of theseestimations include: 1) Establishing a nearly tight approximation result ofDNNs in the Sobolev space; 2) Characterizing the generalization error ofmachine learning methods with loss functions involving function derivatives.This theoretical investigation fills the gap of learning error estimations fora wide range of physics-informed machine learning models and applicationsincluding generative models, solving partial differential equations, operatorlearning, network compression, distillation, regularization, etc.

Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivatives