Physical-Field Reconstruction from Sparse Observations: When Are Diffusion Models Preferable to Deterministic Regression?

Abstract

Reconstructing physical fields from sparse observations is central to system identification, forecasting, and control, yet sparse measurements generally underdetermine the full field. This makes reconstruction an ill-posed inverse problem rather than simple interpolation. Although many deterministic and generative methods have been developed, there is still no clear consensus on when a single point estimate is sufficient and when a distribution of plausible reconstructions is more useful. We conduct a fair comparison of a deterministic U-Net, conditional diffusion, and prior-guided diffusion under matched experimental settings, including 2D Poisson equation, 2D Navier-Stokes flow, and 1D Kuramoto-Sivashinsky dynamics. Through this comparison, we make three observations. First, accuracy is field- and regime-dependent, with no systematic advantage for diffusion under higher complexity or sparser observations. Second, ensemble means improve phase-aligned accuracy, whereas individual samples better preserve variability and can retain high-wavenumber power in selected regimes. Third, conditional diffusion provides more reliable uncertainty estimates at lower cost, while prior-guided diffusion is more robust to mask-distribution shifts but requires substantially higher inference cost and guidance tuning. These results clarify when generative reconstruction is useful and provide guidance for improving uncertainty estimation, fine-scale sample fidelity, robustness, and computational efficiency in sparse field reconstruction.