arXiv:2611.07160v1 Announce Type: new Abstract: Many real-world applications, such as numerical weather prediction or wildfire tracking, involve modeling highly nonlinear dynamical systems that are chaotic and influenced by various sources of uncertainty. To improve model prediction accuracy and reduce such uncertainties, data assimilation (DA) combines the dynamical model with partial, noisy observations to estimate the evolution of the system's state. Doing so depends on solving the optimal filtering problem: computing the conditional distribution of a hidden state given a sequence of observations. Classical approaches fail along three distinct axes: Kalman-based methods assume Gaussian posteriors and produce biased estimates under nonlinear observation operators; particle-based methods represent distributions with weighted samples and collapse in high dimensions due to the curse of dimensionality; and kernel-based methods move beyond discrete particle representations but cannot maintain their density approximation affordably as dimension grows. The Ensemble Score Filter (EnSF) avoids all three constraints by encoding the filtering density in a score function and approximating that score with a training-free Monte Carlo estimator, enabling stable filtering in high-dimensional, chaotic systems. This survey traces EnSF's mathematical foundations, establishes it as the state of the art for non-Gaussian, high-dimensional filtering, and examines fourteen extensions that expand its operational reach. Twelve of the fourteen extensions preserve this training-free Monte Carlo estimator at the core of the update, with only two departing from it to achieve planetary-scale deployment.
Stochastic Dynamics of Generative AI: From Autoregressive Models to Interacting LLM Networks and Neural SDEs