  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.
