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A Priori Neural Networks Versus A Posteriori MOOD Loop: A High Accurate 1D FV Scheme Testing Bed

dc.contributor.authorBOURRIAUD, Alexandre
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorLOUBÈRE, Raphaël
dc.contributor.authorTURPAULT, Rodolphe
dc.date.accessioned2024-04-04T02:47:56Z
dc.date.available2024-04-04T02:47:56Z
dc.date.issued2020-08
dc.identifier.issn0885-7474
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/191692
dc.description.abstractEnIn this work we present an attempt to replace an a posteriori MOOD loop used in a high accurate Finite Volume (FV) scheme by a trained artificial Neural Network (NN). The MOOD loop, by decrementing the reconstruction polynomial degrees, ensures accuracy, essentially non-oscillatory, robustness properties and preserves physical features. Indeed it replaces the classical a priori limiting strategy by an a posteriori troubled cell detection, supplemented with a local time-step re-computation using a lower order FV scheme (ie lower polynomial degree reconstructions). We have trained shallow NNs made of only two so-called hidden layers and few perceptrons which a priori produces an educated guess (classification) of the appropriate polynomial degree to be used in a given cell knowing the physical and numerical states in its vicinity. We present a proof of concept in 1D. The strategy to train and use such NNs is described on several 1D toy models: scalar advection and Burgers' equation, the isentropic Euler and radiative M1 systems. Each toy model brings new difficulties which are enlightened on the obtained numerical solutions. On these toy models, and for the proposed test cases, we observe that an artificial NN can be trained and substituted to the a posteriori MOOD loop in mimicking the numerical admissibility criteria and predicting the appropriate polynomial degree to be employed safely. The physical admissibility criteria is however still dealt with the a posteriori MOOD loop. Constructing a valid training data set is of paramount importance, but once available, the numerical scheme supplemented with NN produces promising results in this 1D setting. Keywords Neural network • Machine learning • Finite Volume scheme • High accuracy • Hyperbolic system • a posteriori MOOD. Mathematics Subject Classification (2010) 65M08 • 65A04 • 65Z05 • 85A25
dc.language.isoen
dc.publisherSpringer Verlag
dc.title.enA Priori Neural Networks Versus A Posteriori MOOD Loop: A High Accurate 1D FV Scheme Testing Bed
dc.typeArticle de revue
dc.identifier.doi10.1007/s10915-020-01282-1
dc.subject.halMathématiques [math]
bordeaux.journalJournal of Scientific Computing
bordeaux.volume84
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue2
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-03084463
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-03084463v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Journal%20of%20Scientific%20Computing&rft.date=2020-08&rft.volume=84&rft.issue=2&rft.eissn=0885-7474&rft.issn=0885-7474&rft.au=BOURRIAUD,%20Alexandre&LOUB%C3%88RE,%20Rapha%C3%ABl&TURPAULT,%20Rodolphe&rft.genre=article


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