GRADIENT-ENHANCED DEEP NEURAL NETWORK APPROXIMATIONS

Abstract

We propose in this work the gradient-enhanced deep neural network (DNN) approach for function approximations and uncertainty quantification. More precisely, the proposed approach adopts both the function evaluations and the associated gradient information to yield enhanced approximation accuracy. In particular, the gradient information is included as a regularization term in the gradient-enhanced DNN approach, for which we present posterior estimates (by the two-layer neural networks) similar to those in the path-norm regularized DNN approximations. We also discuss the application of this approach to gradient-enhanced uncertainty quantification, and present several numerical experiments to show that the proposed approach can outperform the traditional DNN approach in many cases of interest.