Speaker: Deep Ray (University of Maryland)
Title: Learning WENO for entropy stable schemes to solve conservation laws
Abstract: Entropy stable solvers for hyperbolic conservation laws ensure the selection of a physically relevant (weak) solution of the underlying PDE. Among such methods, the TeCNO schemes [Fjordholm et al, 2012] form a class of high-order finite difference-based solvers that utilize reconstruction algorithms satisfying a critical “sign-property” at the cell-interfaces. However, only a handful of existing reconstructions are known to satisfy this property. In [Fjordholm & Ray, 2016], the first weighted essentially non-oscillatory (WENO) reconstruction satisfying the sign- property was developed. However, despite leading to provably entropy stable schemes, the numerical solutions using this reconstruction suffered from large under/overshoots near discontinuities.
In this talk, we propose an alternate approach to constructing WENO schemes possessing the sign-property. In particular, we train a neural network to determine the polynomial weights of the WENO scheme, while strongly constraining the network to satisfy the sign-property. The training data comprises smooth and discontinuous data that represent the local solution features of conservation laws. Additional constraints are built into the network to guarantee the expected order of convergence (for smooth solutions) with mesh refinement. We present several numerical results to demonstrate a significant improvement over the existing variants of WENO with the sign- property.
URL associated with Seminar: https://math.osu.edu/events/tba-computational-math-seminar-deep-ray
