Deep smoothness weighted essentially non-oscillatory method for two-dimensional hyperbolic conservation laws: A deep learning approach for learning smoothness indicators

In this work, we enhance the fifth-order Weighted Essentially Non-Oscillatory (WENO) shock-capturing scheme by integrating deep learning techniques. We improve the established WENO algorithm by training a compact neural network to dynamically adjust the smoothness indicators within the WENO scheme. This modification boosts the accuracy of the numerical results, particularly in proximity to abrupt shocks. Notably, our approach eliminates the need for additional post-processing steps, distinguishing it from previous deep learning-based methods. We substantiate the superiority of our new approach through the examination of multiple examples from the literature concerning the two-dimensional Euler equations of gas dynamics. Through a thorough investigation of these test problems, encompassing various shocks and rarefaction waves, our novel technique consistently outperforms the traditional fifth-order WENO scheme. This superiority is especially evident in cases where numerical solutions exhibit excessive diffusion or overshoot around shocks.