In \cite{wang2023towards}, a dual-consistent dual-weighted residual-based $h$-adaptive method has been proposed based on a Newton-GMG framework, towards the accurate calculation of a given quantity of interest from Euler equations. The performance of such a numerical method is satisfactory, i.e., the stable convergence of the quantity of interest can be observed in all numerical experiments. In this paper, we will focus on the efficiency issue to further develop this method, since efficiency is vital for numerical methods in practical applications such as the optimal design of the vehicle shape. Three approaches are studied for addressing the efficiency issue, i.e., i). using convolutional neural networks as a solver for dual equations, ii). designing an automatic adjustment strategy for the tolerance in the $h$-adaptive process to conduct the local refinement and/or coarsening of mesh grids, and iii). introducing OpenMP, a shared memory parallelization technique, to accelerate the module such as the solution reconstruction in the method. The feasibility of each approach and numerical issues are discussed in depth, and significant acceleration from those approaches in simulations can be observed clearly from a number of numerical experiments. In convolutional neural networks, it is worth mentioning that the dual consistency plays an important role to guarantee the efficiency of the whole method and that unstructured meshes are employed in all simulations.

翻译：在文献[wang2023towards]中，基于牛顿-多重网格框架，针对欧拉方程中给定感兴趣量的精确计算，提出了一种对偶一致的双加权残差h自适应方法。该数值方法的性能令人满意，即在所有数值实验中均可观察到感兴趣量的稳定收敛。鉴于效率对实际应用（如运载器外形优化设计）中的数值方法至关重要，本文将聚焦于效率问题以进一步改进该方法。针对效率问题，本文研究了三种途径：i）利用卷积神经网络作为对偶方程求解器；ii）设计h自适应过程中容差的自动调整策略，实现网格单元的局部加密/粗化；iii）引入共享内存并行化技术OpenMP，加速方法中的解重构等模块。本文深入探讨了各方法的可行性与数值问题，大量数值实验表明，这些方法可显著提升模拟加速效果。值得指出的是，在卷积神经网络中，对偶一致性对保证整个方法的效率具有重要作用，且所有模拟均采用非结构化网格。