WENO3-NN: A maximum-order three-point data-driven weighted essentially non-oscillatory scheme

Neural networks have become more and more relevant for computational fluid dynamics. In recent works, neural network based weighted essentially non-oscillatory schemes have been developed. Challenges faced with such schemes are to ensure maximum-order convergence on narrow stencils and the ENO property. In this work, we use a neural network as a weighting function in the WENO scheme and address these shortcomings. Based on the input stencil, the neural network calculates a convex combination of local interpolation polynomials. We use a Galilean invariant embedding in the input layer and introduce an additional loss on the reconstruction weights, such that the WENO scheme inherently recognizes a smooth input function and achieves maximum-order convergence. The performance of the WENO3-NN scheme is demonstrated for one- and two-dimensional test cases, including strong shocks and shock-density wave interactions. The WENO3-NN scheme shows very good generalizability across all benchmark cases and different resolutions, and exhibits a performance similar to or better than the classical WENO5-JS scheme. By analyzing the approximate dispersion relation of the WENO3-NN scheme, we find that the neural network scheme learns a highly non-trivial dispersion-dissipation relation. Especially, data-driven schemes may introduce vanishing dissipation near the cutoff wavenumber which is counterintuitive to classical discretization-design principles.