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.

翻译：为物理现象建立动力学模型对于理解不同机制间的相互作用以及预测物理状态的演化至关重要。然而，单一动力学模型往往难以胜任这些基础任务，因其存在模型误差与不确定性。常见补救措施是依赖数据同化，通过观测真实系统来更新状态估计。系综滤波器通过随时间更新样本集来顺序同化观测值，其运作分为两步：预测步将每个样本经由动力学模型传播，分析步则利用新观测值更新样本。为对动力系统进行准确鲁棒的预测，离散解必须保留其关键不变量。尽管现代数值求解器能满足这些不变量要求，但现有保持不变量的分析步仅适用于高斯设定，且通常无法与系综滤波器的经典正则化技术（如膨胀和协方差锥化）兼容。本文聚焦于保留线性不变量，如质量、化学物种的化学计量平衡和电荷。利用测度输运理论工具（Spantini等，2022，《SIAM评论》），我们引入一类通用的非线性系综滤波器，能在非高斯滤波问题中自动保留所需线性不变量。将该框架特化至高斯设定时，可恢复卡尔曼滤波的约束形式。随后我们展示了如何将系综卡尔曼滤波（Evensen，1994，《地球物理研究杂志》）的现有正则化技术与线性不变量保留相结合。最后，我们评估了保留线性不变量对系综卡尔曼滤波及非线性系综滤波的益处。