Neural network guided adjoint computations in dual weighted residual error estimation

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Original languageEnglish
Article number62
JournalSN Applied Sciences
Volume4
Issue number2
Early online date31 Jan 2022
Publication statusPublished - Feb 2022

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

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

ASJC Scopus subject areas

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Neural network guided adjoint computations in dual weighted residual error estimation. / Roth, Julian; Schröder, Max; Wick, Thomas.
In: SN Applied Sciences, Vol. 4, No. 2, 62, 02.2022.

Research output: Contribution to journalArticleResearchpeer review

Roth J, Schröder M, Wick T. Neural network guided adjoint computations in dual weighted residual error estimation. SN Applied Sciences. 2022 Feb;4(2):62. Epub 2022 Jan 31. doi: 10.1007/s42452-022-04938-9
Roth, Julian ; Schröder, Max ; Wick, Thomas. / Neural network guided adjoint computations in dual weighted residual error estimation. In: SN Applied Sciences. 2022 ; Vol. 4, No. 2.
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AU - Schröder, Max

AU - Wick, Thomas

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AB - 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.

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