A machine-learning minimal-residual (ML-MRes) framework for goal-oriented finite element discretizations

Brevis, Ignacio; Muga, Ignacio; Zee, Kristoffer G. van der

doi:10.1016/j.camwa.2020.08.012

“…In step 4, we solve the strong form of the adjoint problem, which for nonlinear PDEs or nonlinear goal functionals also depends on the primal solution u l, (2) h . The strong form of the adjoint problem is of the form ( 7) and thus we can find a neural network based solution by minimizing the loss (9) with L-BFGS [29], a quasi-Newton method. We observed that by using L-BFGS sometimes the loss exploded or the optimizer got stuck at a saddle point.…”

Section: Algorithmic Realizationmentioning

confidence: 99%

Neural network guided adjoint computations in dual weighted residual error estimation

Roth¹,

Schröder²,

Wick³

2021

Preprint

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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.

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“…Due to the universal approximation property [42], a primer candidate are neural networks as they are already successfully employed for solving ordinary and partial differential equations (PDE) [6,10,24,25,30,31,33,34,44,45,50,52,58]. A related work in aiming to improve goal-oriented computations with the help of neural network data-driven finite elements is [9]. Moreover, a recent summary of the key concepts of neural networks and deep learning was compiled in [26].…”

Section: Introductionmentioning

confidence: 99%

Neural network guided adjoint computations in dual weighted residual error estimation

Roth

¹

,

Schröder

²

,

Wick

³

2022

View full text Add to dashboard Cite

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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“…

We consider the data-driven acceleration of Galerkin-based finite element discretizations for the approximation of partial differential equations (PDEs) [1]. The aim is to obtain approximations on meshes that are very coarse, but nevertheless resolve quantities of interest with striking accuracy.Our work is inspired by the the machine learning framework of Mishra [2], who considered the data-driven acceleration of finite-difference schemes.

…”

mentioning

confidence: 99%

“…We consider the data-driven acceleration of Galerkin-based finite element discretizations for the approximation of partial differential equations (PDEs) [1]. The aim is to obtain approximations on meshes that are very coarse, but nevertheless resolve quantities of interest with striking accuracy.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Data-Driven Goal-Oriented Finite Element Methods: A~Machine-Learning Minimal-Residual (ML-MRes) Framework

Zee¹,

Brevis²,

Muga³

2021

10th International Conference on Adaptative Modeling and Simulation

0

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We consider the data-driven acceleration of Galerkin-based finite element discretizations for the approximation of partial differential equations (PDEs) [1]. The aim is to obtain approximations on meshes that are very coarse, but nevertheless resolve quantities of interest with striking accuracy.Our work is inspired by the the machine learning framework of Mishra [2], who considered the data-driven acceleration of finite-difference schemes. The essential idea is to optimize a numerical method for a given coarse mesh, by minimizing a loss function consisting of errors with respect to the quantities of interest for obtained training data.

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A machine-learning minimal-residual (ML-MRes) framework for goal-oriented finite element discretizations

Cited by 25 publications

References 36 publications

Neural network guided adjoint computations in dual weighted residual error estimation

Neural network guided adjoint computations in dual weighted residual error estimation

Neural network guided adjoint computations in dual weighted residual error estimation

Data-Driven Goal-Oriented Finite Element Methods: A~Machine-Learning Minimal-Residual (ML-MRes) Framework

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