Neural network guided adjoint computations in dual weighted residual error estimation

TL;DR

This paper introduces a neural network-based approach to solve adjoint problems in goal-oriented error estimation, aiming to reduce computational costs while maintaining accuracy, validated through numerical experiments and software integration.

Contribution

It presents a novel neural network guided method for adjoint computations in error estimation, integrating deep learning with finite element methods for efficiency.

Findings

Neural network adjoint solutions closely match classical finite element solutions.

The approach reduces computational costs for adjoint problems.

Successful implementation with open-source finite element and deep learning libraries.

Abstract

In this work, we are concerned with neural network guided goal-oriented a posteriori error estimation and adaptivity using the dual weighted residual method. The primal problem is solved using classical Galerkin finite elements. The adjoint problem is solved in strong form with a feedforward neural network using two or three hidden layers. The main objective of our approach is to explore alternatives for solving the adjoint problem with greater potential of a numerical cost reduction. The proposed algorithm is based on the general goal-oriented error estimation theorem including both linear and nonlinear stationary partial differential equations and goal functionals. Our developments are substantiated with some numerical experiments that include comparisons of neural network computed adjoints and classical finite element solutions of the adjoints. In the programming software, the open-source library deal.II is successfully coupled with LibTorch, the PyTorch C++ application programming interface.