  In this paper we enhance the well-known fifth order WENO shock-capturing
scheme by using deep learning techniques. This fine-tuning of an existing
algorithm is implemented by training a rather small neural network to modify
the smoothness indicators of the WENO scheme in order to improve the numerical
results especially at discontinuities. In our approach no further
post-processing is needed to ensure the consistency of the method, which
simplifies the method and increases the effect of the neural network. Moreover,
the convergence of the resulting scheme can be theoretically proven.
  We demonstrate our findings with the inviscid Burgers' equation, the
Buckley-Leverett equation and the 1-D Euler equations of gas dynamics. Hereby
we investigate the classical Sod problem and the Lax problem and show that our
novel method outperforms the classical fifth order WENO schemes in simulations
where the numerical solution is too diffusive or tends to overshoot at shocks.

Numerical Analysis;Numerical Analysis;Fluid Dynamics