[Paper Review] Flow field tomography with uncertainty quantification using a Bayesian physics-informed neural network

Joseph P. Molnar, Samuel J. Grauer|arXiv (Cornell University)|Aug 20, 2021

Model Reduction and Neural NetworksPhysics and Astronomy68 references49 citations

TL;DR

This paper introduces a Bayesian physics-informed neural network (B-PINN) for direct 2D flow field tomography reconstruction from sparse line-of-sight (LoS) measurements, embedding the Navier–Stokes and advection–diffusion equations into the loss function. The method achieves superior reconstruction accuracy over state-of-the-art algorithms and enables comprehensive uncertainty quantification (UQ), revealing semi-convergence in noisy data and providing robust posterior distributions for model validation.

ABSTRACT

We report a new approach to flow field tomography that uses the Navier-Stokes and advection-diffusion equations to regularize reconstructions. Tomography is increasingly employed to infer 2D or 3D fluid flow and combustion structures from a series of line-of-sight (LoS) integrated measurements using a wide array of imaging modalities. The high-dimensional flow field is reconstructed from low-dimensional measurements by inverting a projection model that comprises path integrals along each LoS through the region of interest. Regularization techniques are needed to obtain realistic estimates, but current methods rely on truncating an iterative solution or adding a penalty term that is incompatible with the flow physics to varying degrees. Physics-informed neural networks (PINNs) are new tools for inverse analysis that enable regularization of the flow field estimates using the governing physics. We demonstrate how a PINN can be leveraged to reconstruct a 2D flow field from sparse LoS-integrated measurements with no knowledge of the boundary conditions by incorporating the measurement model into the loss function used to train the network. The resulting reconstructions are remarkably superior to reconstructions produced by state-of-the-art algorithms, even when a PINN is used for post-processing. However, as with conventional iterative algorithms, our approach is susceptible to semi-convergence when there is a high level of noise. We address this issue through the use of a Bayesian PINN, which facilitates comprehensive uncertainty quantification of the reconstructions, enables the use of a more intuitive loss function, and reveals the source of semi-convergence.

Motivation & Objective

• . The paper aims to address the ill-posedness and noise sensitivity of traditional flow field tomography reconstructions by integrating physical laws directly into the reconstruction process.
• It seeks to overcome limitations of conventional regularization techniques that are incompatible with fluid dynamics physics.
• The study aims to demonstrate that direct reconstruction using a physics-informed neural network (PINN) yields more accurate results than post-processing standard reconstructions with a PINN.
• It investigates how Bayesian inference within the PINN framework enables uncertainty quantification (UQ) and improves robustness to noise and model errors.
• The objective includes validating the B-PINN's ability to produce reliable posterior distributions for quantitative model comparison and benchmarking.

Proposed method

• . The method uses a physics-informed neural network (PINN) where the loss function incorporates both the measurement model (projection matrix A) and the governing Navier–Stokes and advection–diffusion equations.
• The PINN is trained end-to-end to reconstruct the 2D flow field directly from LoS-integrated measurements, bypassing intermediate reconstruction steps.
• A Bayesian PINN (B-PINN) is employed to sample from the posterior distribution of network weights, enabling full uncertainty quantification (UQ) of the reconstructed fields.
• The B-PINN uses a prior that enforces strict adherence to fluid physics, allowing for more robust optimization and improved reconstructions under noisy conditions.
• The training process is monitored via a three-phase regime: phase I (initial convergence), phase II (physical consistency), and phase III (semi-convergence due to noise).
• A stopping criterion based on the transition from phase I to II is proposed to regularize training and avoid overfitting to noise in real-world data.

Experimental results

Research questions

• RQ1 . Can a PINN be used to directly reconstruct 2D flow fields from sparse LoS measurements by embedding the projection model into the loss function, without relying on a prior reconstruction step?
• RQ2 How does direct PINN reconstruction compare in accuracy to PINN post-processing of standard reconstruction algorithms?
• RQ3 What causes semi-convergence in noisy data reconstructions, and can it be detected and mitigated using a principled training regime?
• RQ4 Can a Bayesian PINN provide more reliable and informative uncertainty estimates than a conventional PINN (MAP estimate) in flow field tomography?
• RQ5 To what extent can a stricter physics prior in the B-PINN improve reconstruction accuracy under noisy conditions?

Key findings

• . Direct reconstruction using a PINN outperformed PINN post-processing of standard reconstructions, even with fewer and noisier projections.
• The method identified three training regimes: initial convergence, physical consistency, and semi-convergence under noise, with the transition from phase I to II serving as a reliable stopping criterion.
• Semi-convergence in noisy data was confirmed via loss function analysis, showing non-convex behavior at low regularization (γ = 10−7), indicating overfitting risk.
• Bayesian PINNs produced conditional mean (CM) reconstructions that were more representative of the true flow field than C-PINN MAP estimates, especially under noise.
• Uncertainty maps from the B-PINN revealed elevated uncertainty in beam gaps and domain edges, aligning with sparse measurement regions and validating the method’s reliability assessment capability.
• Marginal posterior distributions from the B-PINN showed that ground truth values consistently fell within high-posterior-density regions, confirming the method’s robustness and utility for model validation.