Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivatives
It addresses learning error estimation for physics-informed machine learning models and applications like solving PDEs and generative models, providing foundational theoretical insights.
The paper tackles the problem of estimating VC-dimension and pseudo-dimension bounds for derivatives of deep neural networks, achieving nearly optimal results. This enables nearly tight approximation in Sobolev spaces and characterization of generalization errors for methods using derivative-based loss functions.
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.