Details
| Original language | English |
|---|---|
| Article number | 62 |
| Journal | SN Applied Sciences |
| Volume | 4 |
| Issue number | 2 |
| Early online date | 31 Jan 2022 |
| Publication status | Published - 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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In: SN Applied Sciences, Vol. 4, No. 2, 62, 02.2022.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Neural network guided adjoint computations in dual weighted residual error estimation
AU - Roth, Julian
AU - Schröder, Max
AU - Wick, Thomas
N1 - Funding Information: This work is supported by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy within the cluster of Excellence PhoenixD (EXC 2122, Project ID 390833453). Moreover, we thank the anonymous reviewers for several suggestions that helped to improve the paper.
PY - 2022/2
Y1 - 2022/2
N2 - 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.
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.
KW - A posteriori error estimation
KW - Adjoint
KW - Deal.II
KW - Dual weighted residuals
KW - LibTorch
KW - Neural network
UR - http://www.scopus.com/inward/record.url?scp=85123986247&partnerID=8YFLogxK
U2 - 10.1007/s42452-022-04938-9
DO - 10.1007/s42452-022-04938-9
M3 - Article
AN - SCOPUS:85123986247
VL - 4
JO - SN Applied Sciences
JF - SN Applied Sciences
IS - 2
M1 - 62
ER -