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

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Abstract

In 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.

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Notes

  1. We must be careful that the NN may be certain of its prediction, even if it is a wrong one.

  2. Notice that without the a posteriori MOOD loop the 2nd and 4th order schemes crash due to the occurrence of negative densities.

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Acknowledgements

RL would like to thank M. Han Veiga (University of Zürich) and S. Clain (Universidade do Minho) for sharing fruitful discussions on neural networks. The authors thank the anonymous reviewers who have led to an improved version of this paper. The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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  1. Institut de Mathématiques de Bordeaux (IMB), CNRS, Bordeaux INP, Université de Bordeaux, 33400, Talence, France

    Alexandre Bourriaud, Raphaël Loubère & Rodolphe Turpault

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  1. Alexandre Bourriaud
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  2. Raphaël Loubère
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  3. Rodolphe Turpault
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Correspondence to Raphaël Loubère.

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Bourriaud, A., Loubère, R. & Turpault, R. A Priori Neural Networks Versus A Posteriori MOOD Loop: A High Accurate 1D FV Scheme Testing Bed. J Sci Comput 84, 31 (2020). https://doi.org/10.1007/s10915-020-01282-1

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