Physical-field reconstruction from sparse observations: when are diffusion models preferable to deterministic regression?
Year: 2026
Type: article
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 controlled 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 offers a favorable uncertainty–cost trade-off, while prior-guided diffusion accommodates the tested observation-mask shifts without retraining 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.
Authors: Hao Zhou, Rui Zhang, Qi Wang, Hao Sun
Institution: Renmin University of China
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: gold
Award: G8801168398
OpenAlex ID: W7212070602