LoPhyDA: Low-Rank Tensor and Physics Gradient Guided Diffusion for Atmospheric Data Assimilation

Data Assimilation (DA) aims to integrate observations with model forecasts to estimate the state of dynamical systems. Despite the widespread application of diffusion-based assimilation methods, they remain constrained by the high dimensionality of atmospheric states and the reliance on imperfect state-observation mappings. This leaves regions lacking observations spatially unconstrained, leading to severe error accumulation and physical inconsistency.. In this paper, we propose LoPhyDA, a diffusion assimilation algorithm dual-guided by low-rank tensor and physical gradients. By leveraging the low-rank property of meteorological field, we employ tensor completion to exploit spatial continuity and dynamic correlations, reconstructing a globally informative dense field from sparse observations to serve as a global prior. This approach mitigates the information deficit inherent in sparse settings. The framework further incorporates physical constraints into the iterative denoising process, utilizing Partial Differential Equation (PDE) residual gradients to rectify the generative trajectory in real-time. Experimental results demonstrate that LoPhyDA outperforms state-of-the-art generative assimilation models in global weather prediction. It achieves robust and physically consistent assimilation, significantly reducing error accumulation in regions lacking observations.