We formulate and solve a Bayesian inverse Navier-Stokes (N-S) problem that assimilates velocimetry data in order to jointly reconstruct a 3D flow field and learn the unknown N-S parameters, including the boundary position. By hardwiring a generalised N-S problem, and regularising its unknown parameters using Gaussian prior distributions, we learn the most likely parameters in a collapsed search space. The most likely flow field reconstruction is then the N-S solution that corresponds to the learned parameters. We develop the method in the variational setting and use a stabilised Nitsche weak form of the N-S problem that permits the control of all N-S parameters. To regularise the inferred the geometry, we use a viscous signed distance field (vSDF) as an auxiliary variable, which is given as the solution of a viscous Eikonal boundary value problem. We devise an algorithm that solves this inverse problem, and numerically implement it using an adjoint-consistent stabilised cut-cell finite element method. We then use this method to reconstruct magnetic resonance velocimetry (flow-MRI) data of a 3D steady laminar flow through a physical model of an aortic arch for two different Reynolds numbers and signal-to-noise ratio (SNR) levels (low/high). We find that the method can accurately i) reconstruct the low SNR data by filtering out the noise/artefacts and recovering flow features that are obscured by noise, and ii) reproduce the high SNR data without overfitting. Although the framework that we develop applies to 3D steady laminar flows in complex geometries, it readily extends to time-dependent laminar and Reynolds-averaged turbulent flows, as well as non-Newtonian (e.g. viscoelastic) fluids.

翻译：我们构建并求解了一个贝叶斯反演纳维-斯托克斯（N-S）问题，该问题通过同化速度测量数据来联合重建三维流场并学习未知的N-S参数（包括边界位置）。通过硬编码广义N-S问题，并利用高斯先验分布对其未知参数进行正则化，我们在坍缩的搜索空间中学习最可能的参数。随后，最可能的流场重建即为对应所学参数的N-S方程解。我们在变分框架下发展该方法，并采用允许控制所有N-S参数的稳定化Nitsche弱形式。为了正则化推断的几何形状，我们使用粘性符号距离场（vSDF）作为辅助变量，该变量通过求解粘性Eikonal边值问题获得。我们设计了一种求解该反演问题的算法，并采用伴随一致的稳定化切割单元有限元方法进行数值实现。随后，我们运用该方法重建了流经主动脉弓物理模型的三维定常层流磁共振速度成像（flow-MRI）数据，涵盖两种不同雷诺数及信噪比（低/高）水平。研究发现，该方法能够：i）准确重建低信噪比数据，通过滤除噪声/伪影并恢复被噪声掩盖的流动特征；ii）精确复现高信噪比数据且不过度拟合。尽管我们开发的框架适用于复杂几何中的三维定常层流，但其可自然扩展至瞬态层流与雷诺平均湍流，以及非牛顿（如粘弹性）流体。