We present a potent computational method for the solution of inverse problems in fluid mechanics. We consider inverse problems formulated in terms of a deterministic loss function that can accommodate data and regularization terms. We introduce a multigrid decomposition technique that accelerates the convergence of gradient-based methods for optimization problems with parameters on a grid. We incorporate this multigrid technique to the ODIL (Optimizing a DIscrete Loss) framework. The multiresolution ODIL (mODIL) accelerates by an order of magnitude the original formalism and improves the avoidance of local minima. Moreover, mODIL accommodates the use of automatic differentiation for calculating the gradients of the loss function, thus facilitating the implementation of the framework. We demonstrate the capabilities of mODIL on a variety of inverse and flow reconstruction problems: solution reconstruction for the Burgers equation, inferring conductivity from temperature measurements, and inferring the body shape from wake velocity measurements in three dimensions. We also provide a comparative study with the related, popular Physics-Informed Neural Networks (PINNs) method. We demonstrate that mODIL has three to five orders of magnitude lower computational cost than PINNs in benchmark problems including simple PDEs and lid-driven cavity problems. Our results suggest that mODIL is a very potent, fast and consistent method for solving inverse problems in fluid mechanics.

翻译：我们提出了一种用于求解流体力学逆问题的强效计算方法。考虑由确定性损失函数定义的反问题，该损失函数可兼容数据项与正则化项。我们引入一种多重网格分解技术，能够加速基于梯度的网格参数优化方法的收敛速度。将该多重网格技术集成到ODIL（优化离散损失）框架中，形成多分辨率ODIL（mODIL）方法。该方法将原始框架的计算速度提升一个数量级，并增强了避开局部极小值的能力。此外，mODIL支持使用自动微分计算损失函数的梯度，从而简化框架实现过程。我们通过多种逆问题与流动重构案例验证了mODIL的性能：Burgers方程解重构、基于温度测量值反演热导率、以及基于三维尾流速度测量值反演物体形状。我们还与广受欢迎的物理信息神经网络（PINNs）方法进行了对比研究。结果表明，在包含简单偏微分方程和顶盖驱动空腔等基准问题中，mODIL的计算成本比PINNs低三到五个数量级。我们的研究证实，mODIL是一种求解流体力学逆问题的强效、快速且稳定的方法。