Diffusion model-based ensemble generation for highly nonlinear systems with limited ensemble size

We propose a computational strategy that incorporates time-lagged sampling into the reverse process of diffusion-based machine learning models. This approach aims to maintain ensemble diversity and prevent ensemble collapse related to underestimation of uncertainties, particularly in situations with limited ensemble sizes. We demonstrate the method's effectiveness using the Lorenz '96 system as a theoretical testbed, focusing on its highly nonlinear regime without explicitly resolving multiscale interactions. Our approach facilitates stable forecast-analysis cycles at small ensemble sizes without requiring covariance inflation or localization by enhancing the ensemble representation of uncertainties inherent in diffusion models and stochastic dynamics. The proposed models maintain superior temporal coherence. While the probability density function is narrowed under severely undersampled ensembles, stochastic variants recover distributional accuracy as ensemble size increases, whereas deterministic variants show persistent limitations but provide substantial computational efficiency. The method generalizes successfully to systems with increased multiscale complexity and maintains stable long-term performance without drift. Results indicate that active diversity preservation in ensemble generation can enable robust data assimilation in undersampled situations, with potential applications in geophysical flows, plasma systems, and turbulence modeling.