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A Priori Neural Networks Versus A Posteriori MOOD Loop: A High Accurate 1D FV Scheme Testing Bed
Journal of Scientific Computing ( IF 2.5 ) Pub Date : 2020-07-25 , DOI: 10.1007/s10915-020-01282-1
Alexandre Bourriaud , Raphaël Loubère , Rodolphe Turpault

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.



中文翻译:

先验神经网络与后验MOOD回路:高精度一维FV方案测试台

在这项工作中,我们尝试通过训练有素的人工神经网络(NN)来替换高精度有限体积(FV)方案中使用的后验MOOD回路。通过减小重构多项式的度,MOOD循环可确保准确性,基本上无振荡的鲁棒性,并保留物理特征。实际上,它通过后验故障细胞检测代替了传统的先验限制策略,并补充了使用较低阶FV方案(即较低多项式重构)的局部时间步长重新计算。我们训练了仅由两个所谓的隐藏层和少量感知器组成浅层NN,这些感知器是先验的产生已知多项式度的有根据的猜测(分类),以在给定单元格中使用,知道其附近的物理和数值状态。我们提出一维的概念证明。在几种一维玩具模型中描述了训练和使用这种神经网络的策略:标量对流和Burgers方程,等熵欧拉和辐射M1系统。每个玩具模型都带来了新的困难,这些困难对获得的数值解具有启发性。在这些玩具模型上,对于建议的测试用例,我们观察到可以模拟模拟的可接纳性标准并预测要安全使用的适当多项式,可以训练人工NN并替换为后验MOOD循环。但是,实际可接纳性标准仍在处理中。后验MOOD循环。构建有效的训练数据集至关重要,但是一旦可用,以NN为补充的数值方案在此一维设置中将产生令人鼓舞的结果。

更新日期:2020-07-25
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