Volume 73, 2023CEMRACS 2021 - Data Assimilation and Reduced Modeling for High Dimensional Problems
|48 - 67
|30 August 2023
Deep learning-based schemes for singularly perturbed convection-diffusion problems*
Fédération de Mathématiques, CentraleSupélec FR3487 CNRS, Paris-Sacla University, France
2 Ecole des Ponts ParisTech & MATHERIALS INRIA team-project, Marne-la-Vallée, France
3 Cermics, École des Ponts, INRIA, Marne-la-Vallée, France
4 COMMEDIA, INRIA Paris, France & LJLL, Sorbonne Université
5 Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB, Eindhoven, Netherlands
6 LJLL, Sorbonne Université, Paris, France
Deep learning-based numerical schemes such as Physically Informed Neural Networks (PINNs) have recently emerged as an alternative to classical numerical schemes for solving Partial Differential Equations (PDEs). They are very appealing at first sight because implementing vanilla versions of PINNs based on strong residual forms is easy, and neural networks offer very high approximation capabilities. However, when the PDE solutions are low regular, an expert insight is required to build deep learning formulations that do not incur in variational crimes. Optimization solvers are also significantly challenged, and can potentially spoil the final quality of the approximated solution due to the convergence to bad local minima, and bad generalization capabilities. In this paper, we present an exhaustive numerical study of the merits and limitations of these schemes when solutions exhibit low-regularity, and compare performance with respect to more benign cases when solutions are very smooth. As a support for our study, we consider singularly perturbed convection-diffusion problems where the regularity of solutions typically degrades as certain multiscale parameters go to zero.
Mathematics Subject Classification: 68T07 / 65Y20 / 65Y10
Key words: neural networks / pinns / singularly perturbed problems
This project was partly funded by the Emergence project grant of the Paris City Council “Models and Measures”. Funding from I-Site FUTURE is also acknowledged. This publication is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (Grant Agreement n. 810367).
© EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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