# Multigoal-oriented dual-weighted-residual error estimation using deep neural networks

arXiv: https://arxiv.org/abs/2112.11360 · HTML: https://mlsanity.com/paper/2112.11360/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 51 references that are in the map.
- First posted on arXiv 21 Dec 2021.

## Builds on (most cited first, 10 of 10)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications](https://mlsanity.com/paper/1908.10407/) (2019)
- [Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations](https://mlsanity.com/paper/1709.05963/) (2017)
- [Data-driven discovery of PDEs in complex datasets](https://mlsanity.com/paper/1808.10788/) (2018)
- [DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data](https://mlsanity.com/paper/1908.04463/) (2019)
- [Deep Relaxation: partial differential equations for optimizing deep neural networks](https://mlsanity.com/paper/1704.04932/) (2017)
- [A machine learning framework for data driven acceleration of computations of differential equations](https://mlsanity.com/paper/1807.09519/) (2018)
- [Evolution of Novel Activation Functions in Neural Network Training with Applications to Classification of Exoplanets](https://mlsanity.com/paper/1906.01975/) (2019)

## Built on by (most cited first, 2 of 2)

- [Learning robust marking policies for adaptive mesh refinement](https://mlsanity.com/paper/2207.06339/) (2022)
- [E2N: Error Estimation Networks for Goal-Oriented Mesh Adaptation](https://mlsanity.com/paper/2207.11233/) (2022)

## Abstract (arXiv)

Deep learning has shown successful application in visual recognition and certain artificial intelligence tasks. Deep learning is also considered as a powerful tool with high flexibility to approximate functions. In the present work, functions with desired properties are devised to approximate the solutions of PDEs. Our approach is based on a posteriori error estimation in which the adjoint problem is solved for the error localization to formulate an error estimator within the framework of neural network. An efficient and easy to implement algorithm is developed to obtain a posteriori error estimate for multiple goal functionals by employing the dual-weighted residual approach, which is followed by the computation of both primal and adjoint solutions using the neural network. The present study shows that such a data-driven model based learning has superior approximation of quantities of interest even with relatively less training data. The novel algorithmic developments are substantiated with numerical test examples. The advantages of using deep neural network over the shallow neural network are demonstrated and the convergence enhancing techniques are also presented