Structure preservation for the Deep Neural Network Multigrid Solver

Margenberg, Nils; Lessig, Christian; Richter, Thomas

doi:10.1553/etna_vol56s86

Cited by 5 publications

(1 citation statement)

References 38 publications

(27 reference statements)

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“…Nearly exact values are obtained in case of the lift functional (right) but also drag (middle) is getting close to the reference value, in particular for 64 × 3. For the divergence J div the improvement is more modest although compared to our previous results [15] ([14, Fig. 1] in the corresponding preprint) with a network with 32 × 1 GRU cells we still obtain a lower divergence.…”

Section: Accuracy Of the Networkcontrasting

confidence: 56%

Deep neural networks for geometric multigrid methods

Margenberg,

Jendersie,

Richter

et al. 2021

Preprint

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We investigate scaling and efficiency of the deep neural network multigrid method (DNN-MG). DNN-MG is a novel neural network-based technique for the simulation of the Navier-Stokes equations that combines an adaptive geometric multigrid solver, i.e. a highly efficient classical solution scheme, with a recurrent neural network with memory. The neural network replaces in DNN-MG one or multiple finest multigrid layers and provides a correction for the classical solve in the next time step. This leads to little degradation in the solution quality while substantially reducing the overall computational costs. At the same time, the use of the multigrid solver at the coarse scales allows for a compact network that is easy to train, generalizes well, and allows for the incorporation of physical constraints. Previous work on DNN-MG focused on the overall scheme and how to enforce divergence freedom in the solution. In this work, we investigate how the network size affects training and solution quality and the overall runtime of the computations. Our results demonstrate that larger networks are able to capture the flow behavior better while requiring only little additional training time. At runtime, the use of the neural network correction can even reduce the computation time compared to a classical multigrid simulation through a faster convergence of the nonlinear solve that is required at every time step.

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Section: Accuracy Of the Networkcontrasting

confidence: 56%

Deep neural networks for geometric multigrid methods

Margenberg,

Jendersie,

Richter

et al. 2021

Preprint

Self Cite

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Using Deep Neural Networks for Detecting Spurious Oscillations in Discontinuous Galerkin Solutions of Convection-Dominated Convection–Diffusion Equations

Frerichs-Mihov,

Henning,

John

2023

J Sci Comput

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Standard discontinuous Galerkin finite element solutions to convection-dominated convection–diffusion equations usually possess sharp layers but also exhibit large spurious oscillations. Slope limiters are known as a post-processing technique to reduce these unphysical values. This paper studies the application of deep neural networks for detecting mesh cells on which slope limiters should be applied. The networks are trained with data obtained from simulations of a standard benchmark problem with linear finite elements. It is investigated how they perform when applied to discrete solutions obtained with higher order finite elements and to solutions for a different benchmark problem.

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On Loss Functionals for Physics-Informed Neural Networks for Steady-State Convection-Dominated Convection-Diffusion Problems

Frerichs-Mihov,

Henning,

John

2024

Commun. Appl. Math. Comput.

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Solutions of convection-dominated convection-diffusion problems usually possess layers, which are regions where the solution has a steep gradient. It is well known that many classical numerical discretization techniques face difficulties when approximating the solution to these problems. In recent years, physics-informed neural networks (PINNs) for approximating the solution to (initial-)boundary value problems ((I)BVPs) received a lot of interest. This paper studies various loss functionals for PINNs that are especially designed for convection-dominated convection-diffusion problems and that are novel in the context of PINNs. They are numerically compared to the vanilla and an hp-variational loss functional from the literature based on two steady-state benchmark problems whose solutions possess different types of layers. We observe that the best novel loss functionals reduce the $$L^2(\varOmega )$$ L 2 ( Ω ) error by 17.3% for the first and 5.5% for the second problem compared to the methods from the literature.

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Structure preservation for the Deep Neural Network Multigrid Solver

Cited by 5 publications

References 38 publications

Deep neural networks for geometric multigrid methods

Deep neural networks for geometric multigrid methods

Using Deep Neural Networks for Detecting Spurious Oscillations in Discontinuous Galerkin Solutions of Convection-Dominated Convection–Diffusion Equations

On Loss Functionals for Physics-Informed Neural Networks for Steady-State Convection-Dominated Convection-Diffusion Problems

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