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Reduced Basis’ Acquisition by a Learning Process for Rapid On-line Approximation of Solution to PDE’s: Laminar Flow Past a Backstep

  • S.I.: Machine learning in computational mechanics
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Abstract

Reduced basis methods for the approximation to parameter-dependent partial differential equations are now well-developed and start to be used for industrial applications. The classical implementation of the reduced basis method goes through two stages: in the first one, offline and time consuming, from standard approximation methods a reduced basis is constructed; then in a second stage, online and very cheap, a small problem, of the size of the reduced basis, is solved. The offline stage is a learning one from which the online stage can proceed efficiently. In this paper we propose to exploit machine learning procedures in both offline and online stages to either tackle different classes of problems or increase the speed-up during the online stage. The method is presented through a simple flow problem—a flow past a backward step governed by the Navier Stokes equations—which shows, however, interesting features.

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Notes

  1. Note, however, that this does not imply that numerical methods are not worthy as they can lead to new solutions and also are able to be connected to optimization or control algorithms based on mathematical concepts.

  2. From now on, we restrict our attention to the velocity field. The same discussion can be extended to the pressure field.

  3. Remember that this Piola transformation preserves the incompressibility condition satisfied by the velocity field.

  4. Note that in this paper regression trees have been chosen because it is a classical regressor, but any other regressor could have been used.

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Authors and Affiliations

  1. Sorbonne Universités, UPMC Univ. Paris 06 and CNRS, UMR 7606, LIP6, 4, Place Jussieu, Paris Cedex 05, 75252, France

    Patrick Gallinari & Olivier Schwander

  2. Laboratoire Jacques-Louis Lions 4, Sorbonne Universités, UPMC Univ. Paris 06 and CNRS, UMR 7598, Place Jussieu, Paris Cedex 05, 75252, France

    Yvon Maday & Tommaso Taddei

  3. Institut Universtaire de France and Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Yvon Maday

  4. Laboratoire de Statistique Théorique et Appliquée and CNRS, Sorbonne Universités, UPMC Univ. Paris 06, CNRS, UMR 7606, FRE 3684, LIP6, 4, Place Jussieu,75252, Paris Cedex 05, 75252, France

    Maxime Sangnier

Authors
  1. Patrick Gallinari
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  2. Yvon Maday
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  3. Maxime Sangnier
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  4. Olivier Schwander
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  5. Tommaso Taddei
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Correspondence to Yvon Maday.

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All authors declare that they have no conflict of interest.

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This work is part of the activity of the “Institut Carnot Smiles”.

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Gallinari, P., Maday, Y., Sangnier, M. et al. Reduced Basis’ Acquisition by a Learning Process for Rapid On-line Approximation of Solution to PDE’s: Laminar Flow Past a Backstep. Arch Computat Methods Eng 25, 131–141 (2018). https://doi.org/10.1007/s11831-017-9238-z

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  • DOI: https://doi.org/10.1007/s11831-017-9238-z

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