Learning compact shock-capturing schemes from neural networks

A recent work 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. Therefore, 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. 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 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, LD-ROUND achieves higher resolution than classical fifth-order WENO schemes.