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

arXiv: https://arxiv.org/abs/2406.18464 · HTML: https://mlsanity.com/paper/2406.18464/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 74 references that are in the map.
- First posted on arXiv 26 Jun 2024.

## Builds on (most cited first, 7 of 7)

- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Practical Quasi-Newton Methods for Training Deep Neural Networks](https://mlsanity.com/paper/2006.08877/) (2020)
- [Can Physics-Informed Neural Networks beat the Finite Element Method?](https://mlsanity.com/paper/2302.04107/) (2023)
- [Can stable and accurate neural networks be computed? -- On the barriers of deep learning and Smale's 18th problem](https://mlsanity.com/paper/2101.08286/) (2021)
- [4DFlowNet: Super-Resolution 4D Flow MRI using Deep Learning and Computational Fluid Dynamics](https://mlsanity.com/paper/2004.07035/) (2020)
- [Physics-informed compressed sensing for PC-MRI: an inverse Navier-Stokes problem](https://mlsanity.com/paper/2207.01466/) (2022)
- [Joint reconstruction and segmentation of noisy velocity images as an inverse Navier-Stokes problem](https://mlsanity.com/paper/2207.01368/) (2022)

## Built on by (most cited first, 4 of 4)

- [Dynamic Reconstruction of Ultrasound-Derived Flow Fields With Physics-Informed Neural Fields](https://mlsanity.com/paper/2511.01804/) (2025)
- [Learning rheological parameters of non-Newtonian fluids from velocimetry data](https://mlsanity.com/paper/2408.02604/) (2024)
- [Bayesian inference of mean velocity fields and turbulence models from flow MRI](https://mlsanity.com/paper/2412.11266/) (2024)
- [Coupled Reconstruction of 2D Blood Flow and Vessel Geometry from Noisy Images via Physics-Informed Neural Networks and Quasi-Conformal Mapping](https://mlsanity.com/paper/2508.11216/) (2025)

## Abstract (arXiv)

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.