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Description
Many classic methods in data assimilation, like the Ensemble Kalman Filter (EnKF), are limited by its Gaussian ansatz. In this work, we frame the filtering update as learning a nonlinear operator mapping between probability distributions in the mean-field limit. We introduce Measure Neural Mappings (MNMs), a class of neural operators acting on probability measures, implemented via Set Transformers. A central theoretical contribution is establishing the continuum limit for attention mechanisms on measures. We prove that attention layers applied to finite ensembles are consistent with their continuous-measure counterparts, converging in Wasserstein distance as the sample size approaches infinity. This result rigorously links finite-dimensional linear algebra operations to infinite-dimensional operator learning, justifying the use of a single parameterization across different ensemble sizes. We demonstrate that the resulting MNM-enhanced ensemble filter (MNMEF) achieves superior accuracy on chaotic dynamical systems, including the Lorenz '96 and Kuramoto-Sivashinsky models, outperforming leading filtering methods.