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.

翻译：本文研究了深度神经网络（DNNs）导数函数的Vapnik—Chervonenkis维（VC维）与伪维数的近最优估计问题。这两类估计具有两项重要应用：1）在Sobolev空间中建立DNNs的近似紧致性结果；2）刻画涉及函数导数的损失函数下机器学习方法的泛化误差。该理论研究填补了物理信息机器学习模型与应用的泛化误差估计空白，涵盖生成模型、偏微分方程求解、算子学习、网络压缩、知识蒸馏、正则化等广泛领域。