On the learning of high order polynomial reconstructions for essentially non-oscillatory schemes

Approximation accuracy and convergence behavior are essential required properties for the computed numerical solution of differential equations. These requirements restrict the application of deep learning networks in the domain of scientific computing. Moreover, the recipe to create suitable synthetic data which can be used to train a good model is also not very clear. This study focuses on learning of third order essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) reconstructions using classification neural networks with small data sets. In particular, this work (i) proposes a novel way to obtain a third order WENO reconstruction which can be posed as classification problem, (ii) gives simple and novel approach to sample data sets which are small but rich enough to inherit the latent feature of inter-spatial regularity information in the constructed data, (iii) it is established that sampling of train data sets impacts quantitatively as well as qualitatively the required accuracy and non-oscillatory properties of resulting ENO3 and WENO3 schemes, (iv) proposes to use a limiter based multi model to retain desired accuracy as well as non-oscillatory properties of the resulting numerical schemes. Computational results are given which established that learned networks perform well and retain the features of the reconstruction methods.