Learning WENO for entropy stable schemes to solve conservation laws
Philip Charles Deep Ray
Entropy conditions play a crucial role in the extraction of a physically
relevant solution for a system of conservation laws, thus motivating the
construction of entropy stable schemes that satisfy a discrete analogue of such
conditions. TeCNO schemes (Fjordholm et al. 2012) form a class of arbitrary
high-order entropy stable finite difference solvers, which require specialized
reconstruction algorithms satisfying the sign property at each cell interface.
Recently, third-order WENO schemes called SP-WENO (Fjordholm and Ray, 2016) and
SP-WENOc (Ray, 2018) have been designed to satisfy the sign property. However,
these WENO algorithms can perform poorly near shocks, with the numerical
solutions exhibiting large spurious oscillations. In the present work, we
propose a variant of the SP-WENO, termed as Deep Sign-Preserving WENO
(DSP-WENO), where a neural network is trained to learn the WENO weighting
strategy. The sign property and third-order accuracy are strongly imposed in
the algorithm, which constrains the WENO weight selection region to a convex
polygon. Thereafter, a neural network is trained to select the WENO weights
from this convex region with the goal of improving the shock-capturing
capabilities without sacrificing the rate of convergence in smooth regions. The
proposed synergistic approach retains the mathematical framework of the TeCNO
scheme while integrating deep learning to remedy the computational issues of
the WENO-based reconstruction. We present several numerical experiments to
demonstrate the significant improvement with DSP-WENO over the existing
variants of WENO satisfying the sign property.