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

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.

GP-MOOD: A positive-preserving high-order finite volume method for hyperbolic conservation laws

“…We demonstrate that the percentage of those troubled cells, where GP-MOOD's order decrements takes place, is only a fraction of the entire domain and never exceed 10%. Our finding is consistent with the former studies on the polynomial MOOD methods [26,27,34].…”

“…In the worst case, a local solution could end up with the most diffusive -but most stable -solver, e.g., FOG, in the regions where shocks and discontinuities are present. Reportedly, and also will be seen in our results in Section 5, the regions of such troubled cells are only about a few percent (e.g., less than 10% in practice) of the entire domain [26,27,34].…”

“…For this paper to be selfcontained, we provide concise descriptions of those existing concepts while we give sufficient details on new strategies for GP-MOOD. Interested readers refer to [24,25,26,27,28,34] for reviewing the existing MOOD methodologies.…”

“…On the other hand, there has been a new attempt [34] to use a trained Neural Network (NN) with hidden layers and few perceptrons to design an NN-based classification model. In this NN approach, the standard a posterior MOOD detection strategy is replaced by an educated guess of the appropriate order of polynomial accuracy.…”

“…Moreover, the NN-based approach alters the original a posteriori nature of the MOOD paradigm to a trained a priori version, which would be better suited for extending the MOOD method to an implicit scheme. Although promising, the work in [34] has some limitations; namely, it only explored shallow NNs of two hidden layers; investigated 1D tests only; trained their NNs on a relatively moderate size of a training set; the positivity and NAN checks are not in the part of NNs but imposed separately as part of the a posteriori MOOD loop.…”