  Formulating dynamical models for physical phenomena is essential for
understanding the interplay between the different mechanisms and predicting the
evolution of physical states. However, a dynamical model alone is often
insufficient to address these fundamental tasks, as it suffers from model
errors and uncertainties. One common remedy is to rely on data assimilation,
where the state estimate is updated with observations of the true system.
Ensemble filters sequentially assimilate observations by updating a set of
samples over time. They operate in two steps: a forecast step that propagates
each sample through the dynamical model and an analysis step that updates the
samples with incoming observations. For accurate and robust predictions of
dynamical systems, discrete solutions must preserve their critical invariants.
While modern numerical solvers satisfy these invariants, existing
invariant-preserving analysis steps are limited to Gaussian settings and are
often not compatible with classical regularization techniques of ensemble
filters, e.g., inflation and covariance tapering. The present work focuses on
preserving linear invariants, such as mass, stoichiometric balance of chemical
species, and electrical charges. Using tools from measure transport theory
(Spantini et al., 2022, SIAM Review), we introduce a generic class of nonlinear
ensemble filters that automatically preserve desired linear invariants in
non-Gaussian filtering problems. By specializing this framework to the Gaussian
setting, we recover a constrained formulation of the Kalman filter. Then, we
show how to combine existing regularization techniques for the ensemble Kalman
filter (Evensen, 1994, J. Geophys. Res.) with the preservation of the linear
invariants. Finally, we assess the benefits of preserving linear invariants for
the ensemble Kalman filter and nonlinear ensemble filters.
