This paper presents a method for estimating output errors and adapting computational meshes in simulations of unsteady turbulent flows. The chaotic nature of such problems prevents a stable unsteady adjoint solution, and existing regularization techniques are costly for large simulations. The method presented foregoes the unsteady adjoint and instead relies on a field-inversion machine-learning (FIML) framework, which only requires unsteady primal solutions without full-state storage or checkpointing. The FIML model yields an adjoint for the averaged solution, which is combined with an averaged unsteady residual to obtain an output error estimate and adaptive indicator. This error estimate is shown to be accurate when the FIML model augments the original unsteady equations with corrections that are not excessively large. The unsteady residual comes from sampling fine-space residual evaluations during the unsteady simulation. A novel objective function based on an adjoint-weighted residual is presented for the field inversion to improve the ability of the FIML model to predict output errors and the domain-interior state. The localized output error drives adaptation of the mesh size and approximation order. Results for three aerodynamic problems ranging in Reynolds number demonstrate accuracy of the error estimates and efficiency of the computational meshes when compared to other adaptive strategies, including uniform and residual-based refinement.

本文提出了一种在非定常湍流模拟中估计输出误差和调整计算网格的方法。此类问题的混沌性质阻碍了稳定的非定常伴随解和现有的正则化技术对于大型模拟来说成本很高。所提出的方法放弃了不稳定的伴随，而是依赖于场反演机器学习 (FIML) 框架，该框架只需要不稳定的原始解决方案，而无需全状态存储或检查点。FIML 模型为平均解产生伴随，它与平均非定常残差相结合以获得输出误差估计和自适应指标。当 FIML 模型用不太大的修正来增加原始的非定常方程时，这个误差估计被证明是准确的。非定常残差来自在非定常模拟期间对精细空间残差评估进行采样。提出了一种基于伴随加权残差的新目标函数，用于场反演，以提高 FIML 模型预测输出误差和域内部状态的能力。局部输出误差驱动网格大小和近似顺序的适应。与其他自适应策略（包括均匀和基于残差的细化）相比，雷诺数范围内的三个空气动力学问题的结果证明了误差估计的准确性和计算网格的效率。