Guiding diffusion models to reconstruct flow fields from sparse data

TL;DR

This work tackles reconstructing unsteady turbulent flow fields from scarce measurements by coupling diffusion-based generative models with a masked sampling scheme that preserves known data in the reverse process. The authors introduce ConFIG$_u$, a conflict-free, gradient-normalized training strategy that balances diffusion Loss and physics residuals to enforce physical coherence without destabilizing optimization. A masked diffusion sampling method guides the reverse diffusion toward samples consistent with sparse observations, while a smooth Gaussian mask improves continuity around known data points. Across 2D Kolmogorov flow and 3D isotropic turbulence, the approach yields higher distributional fidelity, better physical-consistency metrics, and sharper small-scale structures compared to prior diffusion-based and supervised baselines, using fewer reverse steps. The results underscore diffusion models as a viable, physics-aware tool for high-fidelity flow-field reconstruction from limited measurements with potential applications in CFD and experimental fluid dynamics.

Abstract

The reconstruction of unsteady flow fields from limited measurements is a challenging and crucial task for many engineering applications. Machine learning models are gaining popularity in solving this problem due to their ability to learn complex patterns from data and generalize across diverse conditions. Among these, diffusion models have emerged as particularly powerful in generative tasks, producing high-quality samples by iteratively refining noisy inputs. In contrast to other methods, these generative models are capable of reconstructing the smallest scales of the fluid spectrum. In this work, we introduce a novel sampling method for diffusion models that enables the reconstruction of high-fidelity samples by guiding the reverse process using the available sparse data. Moreover, we enhance the reconstructions with available physics knowledge using a conflict-free update method during training. To evaluate the effectiveness of our method, we conduct experiments on 2 and 3-dimensional turbulent flow data. Our method consistently outperforms other diffusion-based methods in predicting the fluid's structure and in pixel-wise accuracy. This study underscores the remarkable potential of diffusion models in reconstructing flow field data, paving the way for their application in Computational Fluid Dynamics research.

Figures (17)

• Figure 1: a) Sparse data $\mathbf{x}_s$ marked with white crosses over ground truth. b) Smooth Gaussian mask $m^s$.
• Figure 2: Sketch of our proposed masked diffusion method, where a smooth mask is applied to improve the prediction of $\mathbf{x}_0$ during the reverse process.
• Figure 3: A comparison of ConFIG and $\text{ConFIG}_\text{u}$ methods, where $\boldsymbol{g}_F$ is the gradient of the physical loss and $\boldsymbol{g}_D$ the gradient of the diffusion loss.
• Figure 4: Datasets used: (a) 2D Kolmogorov dataset, (b) 3D isotropic turbulence.
• Figure 5: Sigma value exploration for the 2D Kolmogorov case. The red vertical line indicates the chosen value for the parameter. All metrics are normalised.