Deep smoothness WENO scheme for two-dimensional hyperbolic conservation
  laws: A deep learning approach for learning smoothness indicators
Tatiana Kossaczká Ameya D. Jagtap Matthias Ehrhardt
In this paper, we introduce an improved version of the fifth-order weighted
essentially non-oscillatory (WENO) shock-capturing scheme by incorporating deep
learning techniques. The established WENO algorithm is improved by training a
compact neural network to adjust the smoothness indicators within the WENO
scheme. 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 from
the literature for the two-dimensional Euler equations of gas dynamics. Through
intensive study of these test problems, which involve various shocks and
rarefaction waves, the new technique is shown to outperform traditional
fifth-order WENO schemes, especially in cases where the numerical solutions
exhibit excessive diffusion or overshoot around shocks.