Neural network guided adjoint computations in dual weighted residual error estimation

Abstract

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. Article Highlights Adjoint approximation with feedforward neural network in dual-weighted residual error estimation. Side-by-side comparisons for accuracy and computational cost with classical finite element computations. Numerical experiments for linear and nonlinear problems yielding excellent effectivity indices.

Keywords

• Dual weighted residuals
• A posteriori error estimation
• Adjoint
• Neural network
• Deal.II
• LibTorch