Generation of Error Indicators for Partial Differential Equations by Machine Learning Methods
Publiceringsår
2022
Upphovspersoner
Muzalevskiy, Alexey; Neittaanmäki, Pekka; Repin, Sergey
Abstrakt
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.
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Publikationstyp
Publikationsform
Artikel
Moderpublikationens typ
Samlingsverk
Artikelstyp
Annan artikel
Målgrupp
VetenskapligKollegialt utvärderad
Kollegialt utvärderadUKM:s publikationstyp
A3 Del av bok eller annat samlingsverkPublikationskanalens uppgifter
Moderpublikationens namn
Moderpublikationens redaktörer
Tuovinen, Tero T.; Periaux, Jacques; Neittaanmäki, Pekka
Förläggare
Sidor
63-96
ISSN
ISBN
Publikationsforum
Publikationsforumsnivå
2
Öppen tillgång
Öppen tillgänglighet i förläggarens tjänst
Nej
Parallellsparad
Ja
Övriga uppgifter
Vetenskapsområden
Matematik; Data- och informationsvetenskap
Nyckelord
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Publiceringsland
Schweiz
Förlagets internationalitet
Internationell
Språk
engelska
Internationell sampublikation
Ja
Sampublikation med ett företag
Nej
DOI
10.1007/978-3-030-70787-3_6
Publikationen ingår i undervisnings- och kulturministeriets datainsamling
Ja