# Gradient-enhanced deep neural network approximations

arXiv: https://arxiv.org/abs/2211.04226 · HTML: https://mlsanity.com/paper/2211.04226/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 29 references that are in the map.
- First posted on arXiv 8 Nov 2022.

## Builds on (most cited first, 9 of 9)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [A Survey on Deep Learning in Medical Image Analysis](https://mlsanity.com/paper/1702.05747/) (2017)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [A vector-contraction inequality for Rademacher complexities](https://mlsanity.com/paper/1605.00251/) (2016)
- [Random Features for Kernel Approximation: A Survey on Algorithms, Theory, and Beyond](https://mlsanity.com/paper/2004.11154/) (2020)
- [A Priori Estimates of the Population Risk for Two-layer Neural Networks](https://mlsanity.com/paper/1810.06397/) (2018)

## Built on by (most cited first, 3 of 3)

- [CS4ML: A general framework for active learning with arbitrary data based on Christoffel functions](https://mlsanity.com/paper/2306.00945/) (2023)
- [Optimal sampling for least-squares approximation](https://mlsanity.com/paper/2409.02342/) (2024)
- [Sobolev neural network with residual weighting as a surrogate in linear and non-linear mechanics](https://mlsanity.com/paper/2407.16466/) (2024)

## Abstract (arXiv)

We propose in this work the gradient-enhanced deep neural networks (DNNs) 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 DNNs approach, for which we present similar posterior estimates (by the two-layer neural networks) as those in the path-norm regularized DNNs 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 DNNs approach in many cases of interests.