A3 Book section, Chapters in research books
Generation of Error Indicators for Partial Differential Equations by Machine Learning Methods (2022)

Muzalevskiy, A., Neittaanmäki, P., & Repin, S. (2022). Generation of Error Indicators for Partial Differential Equations by Machine Learning Methods. In T. T. Tuovinen, J. Periaux, & P. Neittaanmäki (Eds.), Computational Sciences and Artificial Intelligence in Industry : New Digital Technologies for Solving Future Societal and Economical Challenges (pp. 63-96). Springer. Intelligent Systems, Control and Automation: Science and Engineering, 76. https://doi.org/10.1007/978-3-030-70787-3_6

JYU authors or editors

Publication details

All authors or editors: Muzalevskiy, Alexey; Neittaanmäki, Pekka; Repin, Sergey

Parent publication: Computational Sciences and Artificial Intelligence in Industry : New Digital Technologies for Solving Future Societal and Economical Challenges

Parent publication editors: Tuovinen, Tero T.; Periaux, Jacques; Neittaanmäki, Pekka

ISBN: 978-3-030-70786-6

eISBN: 978-3-030-70787-3

Journal or series: Intelligent Systems, Control and Automation: Science and Engineering

ISSN: 2213-8986

eISSN: 2213-8994

Publication year: 2022

Number in series: 76

Pages range: 63-96

Number of pages in the book: 275

Publisher: Springer

Place of Publication: Cham

Publication country: Switzerland

Publication language: English

DOI: https://doi.org/10.1007/978-3-030-70787-3_6

Publication open access: Not open

Publication channel open access:

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/84830

Additional information: The CSAI 2019 Conference (Computational Science and AI in Industry: New Digital Technologies for Solving Future Societal and Economical Challenges) took place at Jyväskylä, Finland, on June 12–14, 2019.

Abstract

Computer simulation methods for models based on partial differential equations usually apply adaptive strategies that generate sequences of approximations for consequently refined meshes. In this process, error indicators play a crucial role because a new (refined) mesh is created by analysis of an approximate solution computed for the previous (coarser) mesh. Different error indicators exploit various analytical and heuristic arguments. The main goal of this paper is to show that effective indicators of approximation errors can be created by machine learning methods and presented by relatively simple networks. We use the “supervised learning” conception where sequences of teaching examples are constructed due to earlier developed tools of a posteriori error analysis known as “functional type error majorants”. Insensitivity to specific features of approximations is an important property of error majorants, which allows us to generate arbitrarily long series of diverse training examples without restrictions on the type of approximate solutions. These new (network) error indicators are compared with known indicators. The results show that after a proper machine learning procedure, we obtain a network with the same (or even better) quality of error indication level as the most efficient indicators used in classical computer simulation methods. The final trained network is approximately as effective as the gradient averaging error indicator, but has an important advantage because it is valid for a much wider set of approximate solutions.

Keywords: mathematical models; simulation; partial differential equations; approximation; error analysis; machine learning

Free keywords: mesh adaptive methods; machine learning; network error indicator

Contributing organizations

Ministry reporting: Yes

Reporting Year: 2022

Preliminary JUFO rating: 2