Bayesian inverse Navier-Stokes problems: joint flow field reconstruction and parameter learning

TL;DR

This work develops a Bayesian inverse Navier–Stokes framework that jointly reconstructs a 3D flow field and learns unknown NS parameters, including boundary geometry, by assimilating velocimetry data. It combines a variational formulation with a stabilised Nitsche weak form and a viscous signed distance field (vSDF) to represent geometry, solved via an adjoint-based saddle-point solver and a BFGS-based posterior-covariance approximation. The approach is implemented with a meshless fictitious-domain cut-cell finite element method and demonstrated on flow-MRI data of steady laminar flow in a 3D aortic arch at two Reynolds numbers and two SNR levels, achieving noise filtering at low SNR and faithful data-fitting at high SNR without overfitting. Uncertainty quantification is enabled through eigen-decomposition of the posterior covariance, and the NS unknowns remain physically interpretable, allowing extraction of derived quantities like pressure and wall shear stress. The framework generalizes to time-dependent and non-Newtonian flows and supports digital-twin cardiovascular modelling with potential extensions in adaptive discretisation and experimental design.

Abstract

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.

Figures (11)

• Figure 1: When $\mathcal{Z}$ is linear (left), $\pi(\bm{x}|\bm{u}^\star)$ is Gaussian. When $\mathcal{Z}$ is weakly nonlinear (middle), $\pi(\bm{x}|\bm{u}^\star)$ can be approximated reasonably well by a Gaussian p.d.f., $\widetilde{\pi}(\bm{x}|\bm{u}^\star)$, around $\bm{x}^\circ$ (Laplace approximation). When $\mathcal{Z}$ is strongly nonlinear (right), $\pi(\bm{x}|\bm{u}^\star)$ can even be multimodal, in which case there are multiple critical points which will yield different approximations $\widetilde{\pi}(\bm{x}|\bm{u}^\star)$ (even locally, the approximation may be inaccurate).
• Figure 2: The Bayesian inverse N--S problem assimilates velocimetry data, $\bm{u}^\star$, in order to jointly reconstruct the flow field, $\bm{u}$, and learn the unknown N--S parameters, $\bm{x}$. The MAP estimator of the unknown N--S parameters, $\bm{x}^\circ$, is found by solving problem \ref{['eq:inv_prob_aug']}, and the reconstructed velocity field is given by $\bm{u}^\circ=\mathcal{Q}\bm{x}^\circ$.
• Figure 3: The projection operator $\mathcal{S}: \bm{M}\to\bm{D}$ is used to project the modelled velocity field, $\bm{u} \in \bm{M}$, to $\mathcal{S}\bm{u}\in\bm{D}$, so that it can be compared with the velocity data, $\bm{u}^\star$. A mask, $\chi$, is commonly used after pre-processing the data to remove irregularities such as outliers (e.g. defective pixels), or regions with no signal. The adjoint operator $\mathcal{S}^*$ is used to project from $\bm{D}$ to $\bm{M}$ (the inverse of $\mathcal{S}$ does not exist).
• Figure 4: The geometry, $\Omega \subset I_m$, is implicitly defined using a viscous signed distance field (vSDF), ${\varphi_{\pm}} \in L^2(I_m)$, which generates an extension of the unit normal vector on $\partial\Omega$ to the whole domain, $I_m$. The shape gradient, $\zeta \in L^2(\Gamma)$, is then extended to the whole domain, $I_m$, along the ${\mathring{\bm{\nu}}}$-streamlines, by solving a convection-diffusion problem. This produces a perturbation of the vSDF, ${\varphi_\pm'} \in L^2(I_m)$, which models the flow of geometry. The artificial viscosity coefficient, $\varepsilon_{\varphi_{\pm}}$, controls the regularity of the shape $\partial\Omega$ by dissipating small-scale features when assimilating noisy velocimetry data, $\bm{u}^\star$.
• Figure 5: In cut-cell FEM the boundary, $\Gamma$, is implicitly defined by the zeroth level-set of ${\varphi_{\pm}}$. The discretised boundary, $\Gamma_h$, consists of the facets of the cut-cells. Under the assumption of a geometry-resolving mesh, there are only a few different cut-cell types (polyhedra, $\mathcal{P}$, formed by cuboid-plane intersections).