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 (PINN) method. We demonstrate that mODIL provides 200x speedup in terms of iteration number on the lid-driven cavity problem and has orders of magnitude lower computational cost. Our results suggest that mODIL is the fastest and most accurate method for solving 2D and 3D inverse problems in fluid mechanics.

翻译：我们提出了一种求解流体力学反问题的高效计算方法。我们考虑基于确定性损失函数（可包含数据项与正则化项）构建的反问题。引入一种多网格分解技术，可加速基于梯度的优化方法在网格参数问题中的收敛速度。将该多网格技术融入ODIL（优化离散损失）框架。多分辨率ODIL（mODIL）将原始计算速度提升一个量级，并增强了对局部极小值的规避能力。此外，mODIL支持利用自动微分计算损失函数的梯度，从而简化了框架的实现。我们通过多种反问题与流场重建案例验证了mODIL的性能：Burgers方程的解重构、根据温度测量值反演热导率、以及根据三维尾流速度测量值反演物体形状。我们还与流行的物理信息神经网络方法进行了对比研究。结果表明，在顶盖驱动方腔流问题中，mODIL在迭代次数上实现200倍加速，计算成本降低数个数量级。我们的研究表明，mODIL是求解二维与三维流体力学反问题中速度最快、精度最高的方法。