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.

我们提出并解决了一个贝叶斯逆纳维 - 斯托克斯（N-S）问题，通过同化流速测量数据来共同重建 3D 流场并学习未知的 N-S 参数，包括边界位置。我们在一个推理问题中使用通用的 N-S 问题，并使用高斯先验分布正则化其未知参数，以在一个简化的搜索空间中学习最可能的参数。最可能的流场重建是与学习的参数相对应的 N-S 解。我们在变分设置下开发了该方法，并使用 N-S 问题的一个稳定的 Nitsche schwach form，以控制所有的 N-S 参数。为了正则化推断出的几何形状，我们使用粘性符号距离场（vSDF）作为辅助变量，它是作为粘性阻尼 Eikonal 边界值问题的解给出的。我们设计了一个解决这个逆问题的算法，并使用一个相邻一致稳定的截取单元有限元方法在数值上实现它。然后，我们使用这种方法来重建通过主动脉弓物理模型的 3D 稳态层流的磁共振流速测量（流 - MRI）数据，包括两个不同雷诺数和信噪比（低 / 高）。我们发现该方法能够准确地 i）通过滤除噪声 / 伪影并恢复被噪声遮蔽的流动特征来重建低信噪比数据，并且 ii）能够在不过度拟合的情况下重现高信噪比数据。尽管我们开发的框架适用于复杂几何形状中的 3D 稳态层流，但它很容易扩展到时间相关的层流和雷诺平均的湍流流动，以及非牛顿流体（例如粘弹性流体）。