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：我们提出了一种解决流体力学逆差问题的强有力的计算方法。我们考虑了在确定性损失功能方面形成的、能够满足数据和正规化条件的反向问题。我们采用了一种多格分解技术,加速了基于梯度的优化方法与网格参数的趋同。我们将这种多格技术纳入ODIL(优化溶液流失)框架。多分辨率ODIL(MODIL)以原有形式化的规模顺序加速,并改进了对当地微型网络的避免。此外,MODIIL在计算损失函数的梯度时采用了自动区分,从而便利了框架的执行。我们展示了 mODIL在各种反向和流动重建问题上的能力:汉堡方方程式的解决方案重建,从温度测量中推断出导力性能,以及体形形状从三个维度测速度测量中推导出。我们还提供了与相关、流行性物理-内建网络(PINN)网络(PINUD)的对比性研究。此外,MODIL在计算速度方法中显示,MODIL(MIL) 3的精确度和计算方法中,以最快速度计算方法显示,其速度为2的计算结果为2。</s>