The ensemble Kalman filter (EnKF) is widely used for data assimilation in high-dimensional systems, but its performance often deteriorates for strongly nonlinear dynamics due to the structural mismatch between the Kalman update and the underlying system behavior. In this work, we propose a latent autoencoder ensemble Kalman filter (LAE-EnKF) that addresses this limitation by reformulating the assimilation problem in a learned latent space with linear and stable dynamics. The proposed method learns a nonlinear encoder--decoder together with a stable linear latent evolution operator and a consistent latent observation mapping, yielding a closed linear state-space model in the latent coordinates. This construction restores compatibility with the Kalman filtering framework and allows both forecast and analysis steps to be carried out entirely in the latent space. Compared with existing autoencoder-based and latent assimilation approaches that rely on unconstrained nonlinear latent dynamics, the proposed formulation emphasizes structural consistency, stability, and interpretability. We provide a theoretical analysis of learning linear dynamics on low-dimensional manifolds and establish generalization error bounds for the proposed latent model. Numerical experiments on representative nonlinear and chaotic systems demonstrate that the LAE-EnKF yields more accurate and stable assimilation than the standard EnKF and related latent-space methods, while maintaining comparable computational cost and data-driven.

翻译：集成卡尔曼滤波(EnKF)广泛应用于高维系统的数据同化，但针对强非线性动力学，其性能常因卡尔曼更新与底层系统行为之间的结构失配而恶化。本文提出潜自编码器集成卡尔曼滤波(LAE-EnKF)，通过在具有线性稳定动力学的学习潜空间中重新表述同化问题来解决这一局限。该方法联合学习非线性编码器-解码器、稳定的线性潜演化算子及一致的潜观测映射，在潜坐标中构建闭合线性状态空间模型。该构造恢复与卡尔曼滤波框架的兼容性，使预测和分析步骤完全在潜空间中执行。与依赖无约束非线性潜动力学的现有自编码器和潜同化方法相比，本方法强调结构一致性、稳定性和可解释性。我们从理论上分析低维流形上的线性动力学学习，并建立所提潜模型的泛化误差界。在代表性非线性与混沌系统上的数值实验表明：LAE-EnKF在保持相当计算开销与数据驱动特性的同时，相比标准EnKF及相关潜空间方法能实现更精准稳定的同化。