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In this paper, we introduce an improved version of the fifth-order weightedessentially non-oscillatory (WENO) shock-capturing scheme by incorporating deeplearning techniques. The established WENO algorithm is improved by training acompact neural network to adjust the smoothness indicators within the WENOscheme. This modification enhances the accuracy of the numerical results,particularly near abrupt shocks. Unlike previous deep learning-based methods,no additional post-processing steps are necessary for maintaining consistency.We demonstrate the superiority of our new approach using several examples fromthe literature for the two-dimensional Euler equations of gas dynamics. Throughintensive study of these test problems, which involve various shocks andrarefaction waves, the new technique is shown to outperform traditionalfifth-order WENO schemes, especially in cases where the numerical solutionsexhibit excessive diffusion or overshoot around shocks.

Deep smoothness WENO scheme for two-dimensional hyperbolic conservation laws: A deep learning approach for learning smoothness indicators