A unified framework for non-linear reconstruction schemes in a compact stencil. Part 2: Learning operators from neural networks

In the preceding part, a family of ROUND (Reconstruction Operators on Unified Normalised-variable Diagram) schemes was introduced within a unified framework that includes existing non-linear reconstruction schemes developed in a compact stencil. While the initial ROUND schemes demonstrated promising performance, there remains potential to enhance their spectral properties by designing alternative functions within this framework. Recently, Bezgin et al. [1] proposed the WENO3-NN scheme, which employs neural networks as weighting functions in Weighted Essentially Non-Oscillatory (WENO) schemes. Although WENO3-NN exhibits improved accuracy and reduced numerical dissipation, it incurs higher computational costs compared to conventional schemes. In this study, we first unify the WENO3-NN scheme within the normalised-variable diagram framework. This analysis reveals that WENO3-NN deviates from the second-order upwind scheme and tends toward the second-order central scheme as the normalised variable decreases below zero. For less smooth functions, the cell-face reconstruction loss dominates over the ideal weight loss, driving WENO3-NN toward central behaviour. Motivated by this insight, we propose a new ROUND formulation that mirrors the behaviour of WENO3-NN while including a greater contribution from the central scheme to further enhance the dissipation property. The resulting ROUND scheme preserves the key features of WENO3-NN on the normalised-variable diagram while maintaining algorithmic simplicity. Building on this formulation, we further develop a low-dissipative ROUND (LD-ROUND) scheme by introducing controlled anti-dissipation errors. Comparative study of numerical error versus CPU cost demonstrates that both ROUND and LD-ROUND achieve substantial efficiency gains over WENO3-NN. For example, using the same grid, LD-ROUND produces numerical errors and CPU costs that are both less than half of those generated by WENO3-NN. Finally, we validate the proposed schemes on a range of compressible single- and multi-phase flow problems, demonstrating their superior performance. In several benchmark tests, the LD-ROUND scheme achieves higher resolution than classical fifth-order WENO schemes.