Generative Forecasting from Coarse-Grained Observations: When Does Cascading Help?
We study probabilistic forecasting of dynamical systems such as fluids and waves from coarse observations. Since the unseen fine scales influence the dynamics, the same coarse initial state can evolve into a whole distribution of futures. Generative forecasters may produce visually plausible samples, but it is hard to tell whether they sample the right conditional distribution, since trajectory data contain a single future for each initial state. We address this problem by generating reference data in a hierarchical way: for each coarse initial state, we generate many different fine-scale components and numerically evolve the full states. Using these conditional ensembles, we show that standard diffusion models and stochastic interpolants make substantial distributional errors. We then show that cascading shorter forecasts greatly reduces these errors for the turbulent flows and waves we study, but we also find and characterize cases in which it hurts. Since the coarse process is not Markov, cascading discards information about the unresolved fine scales even with exact transitions. To understand this, we propose a solvable analytic model which shows that the optimal cascade depth decreases as the coupling between the unresolved and observed scales gets stronger, and increases as the dependence of the forecast on the observation becomes more oscillatory. Consistent with our experiments, cascading is detrimental when the direct forecast is ``easy'' to learn, for example Gaussian as in the passive scalar case. We release our datasets as a first benchmark for probabilistic forecasting with ground-truth conditional ensembles.