Ensemble-Conditional Gaussian Processes (Ens-CGP): Representation, Geometry, and Inference

We formulate Ensemble-Conditional Gaussian Processes (Ens-CGP), a finite-dimensional synthesis that centers ensemble-based inference on the conditional Gaussian law. Conditional Gaussian processes (CGP) arise directly from Gaussian processes under conditioning and, in linear-Gaussian settings, define the full posterior distribution for a Gaussian prior and linear observations. Classical Kalman filtering is a recursive algorithm that computes this same conditional law under dynamical assumptions; the conditional Gaussian law itself is therefore the underlying representational object, while the filter is one computational realization. In this sense, CGP provides the probabilistic foundation for Kalman-type methods as well as equivalent formulations as a strictly convex quadratic program (MAP estimation), RKHS-regularized regression, and classical regularization. Ens-CGP is the ensemble instantiation of this object, obtained by treating empirical ensemble moments as a (possibly low-rank) Gaussian prior and performing exact conditioning. By separating representation (GP -> CGP -> Ens-CGP) from computation (Kalman filters, EnKF variants, and iterative ensemble schemes), the framework links an earlier-established representational foundation for inference to ensemble-derived priors and clarifies the relationships among probabilistic, variational, and ensemble perspectives.

Learning a Stochastic Differential Equation Model of Tropical Cyclone Intensification from Reanalysis and Observational Data

Tropical cyclones are dangerous natural hazards, but their hazard is challenging to quantify directly from historical datasets due to limited dataset size and quality. Models of cyclone intensification fill this data gap by simulating huge ensembles of synthetic hurricanes based on estimates of the storm's large scale environment. Both physics-based and statistical/ML intensification models have been developed to tackle this problem, but an open question is: can a physically reasonable and simple physics-style differential equation model of intensification be learned from data? In this paper, we answer this question in the affirmative by presenting a 10-term cubic stochastic differential equation model of Tropical Cyclone intensification. The model depends on a well-vetted suite of engineered environmental features known to drive intensification and is trained using a high quality dataset of hurricane intensity (IBTrACS) with estimates of the cyclone's large scale environment from a data-assimilated simulation (ERA5 reanalysis), restricted to the Northern Hemisphere. The model generates synthetic intensity series which capture many aspects of historical intensification statistics and hazard estimates in the Northern Hemisphere. Our results show promise that interpretable, physics style models of complex earth system dynamics can be learned using automated system identification techniques.