An Optimal Control Deep Learning Method to Design Artificial Viscosities for Discontinuous Galerkin Schemes

In this paper, we propose a method for constructing a neural network viscosity in order to reduce the non-physical oscillations generated by high-order Discontinuous Galerkin methods on uniform Cartesian grids. To this end, the problem is reformulated as an optimal control problem for which the cont...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 101; no. 3; p. 70
Main Authors: Bois, Léo, Franck, Emmanuel, Navoret, Laurent, Vigon, Vincent
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2024
Springer Nature B.V
Springer Verlag
Subjects:
Algorithms
Approximation
Back propagation networks
Computational Mathematics and Numerical Analysis
Cost function
Deep learning
Design
Galerkin method
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Methods
Neural networks
Optimal control
Optimization
Partial differential equations
Performance evaluation
Theoretical
Viscosity
Artificial viscosity
Neural networks
Back-propagation
Machine learning
Discontinuous galerkin
machine learning
hyperbolic
high order
ISSN:0885-7474, 1573-7691
Online Access:Get full text
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Summary:In this paper, we propose a method for constructing a neural network viscosity in order to reduce the non-physical oscillations generated by high-order Discontinuous Galerkin methods on uniform Cartesian grids. To this end, the problem is reformulated as an optimal control problem for which the control is the viscosity function and the cost function involves comparison with a reference solution after several compositions of the scheme. The learning process is strongly based on gradient backpropagation tools. Numerical simulations show that the artificial viscosities, with a convolutional architecture, constructed in this way are just as good or better than those used in the literature.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-024-02698-9