This dissertation advances the mathematical and algorithmic foundations of data assimilation (DA) for inference in nonlinear stochastic dynamical systems. DA is a cornerstone of modern computational science, enabling the integration of observational data with dynamical models for state estimation and uncertainty quantification. Classical algorithmic approaches, such as the ensemble Kalman filter (EnKF), rely on Gaussian approximations to estimate the conditional distribution of state given observations, which limit their fidelity in scientific and engineering applications. The first part of this work develops a probabilistic mean-field formulation of the filtering problem, providing a rigorous description of the evolution of the conditional distribution of state given observations. This framework leads to new analytical tools to quantify the uncertainty represented by ensemble filters, culminating in the first proof of the accuracy of the EnKF as a quantifier of uncertainty for near-linear dynamics. The second part develops theoretical and methodological advances in scientific machine learning. A continuum formulation of the attention mechanism enables the design of transformer neural operators together with the first universal approximation theorem for this class of architectures. This line of work further leads to measure neural mappings, the first neural operators defined on spaces of probability measures, opening a new avenue for Bayesian inference. With this novel methodology we introduce two new strategies for enhancing data assimilation via operator learning: (i) a fully data-driven approach involving direct approximation of conditional states via operator learning and (ii) a model-driven approach involving learning the DA analysis map via measure neural mappings. Together, these results pave the way towards theoretical and computational foundations for accurate uncertainty quantification in nonlinear, high-dimensional systems, with implications for climate science, engineering design, and digital twins.
Edoardo Calvello (Tue,) studied this question.