Filtering is a data assimilation technique that performs the sequential inference of dynamical systems states from noisy observations. Herein, we propose a machine learning-based ensemble conditional mean filter (ML-EnCMF) for tracking possibly high-dimensional non-Gaussian state models with nonlinear dynamics based on sparse observations. The proposed filtering method is developed based on the conditional expectation and numerically implemented using machine learning (ML) techniques combined with the ensemble method. The contribution of this work is twofold. First, we demonstrate that the ensembles assimilated using the ensemble conditional mean filter (EnCMF) provide an unbiased estimator of the Bayesian posterior mean, and their variance matches the expected conditional variance. Second, we implement the EnCMF using artificial neural networks, which have a significant advantage in representing nonlinear functions over high-dimensional domains such as the conditional mean. Finally, we demonstrate the effectiveness of the ML-EnCMF for tracking the states of Lorenz-63 and Lorenz-96 systems under the chaotic regime. Numerical results show that the ML-EnCMF outperforms the ensemble Kalman filter.

中文翻译：

---

基于机器学习的条件均值滤波器：用于非线性数据同化的集成卡尔曼滤波器的推广

过滤是一种数据同化技术，它从嘈杂的观察中执行动态系统状态的顺序推断。在此，我们提出了一种基于机器学习的集成条件均值滤波器（ML-EnCMF），用于基于稀疏观察跟踪具有非线性动力学的可能高维非高斯状态模型。所提出的过滤方法是基于条件期望开发的，并使用机器学习 (ML) 技术结合集成方法进行数值实现。这项工作的贡献是双重的。首先，我们证明了使用集成条件均值滤波器 (EnCMF) 同化的集成提供了贝叶斯后验均值的无偏估计量，并且它们的方差与预期的条件方差相匹配。其次，我们使用人工神经网络实现 EnCMF，它在表示非线性函数方面比高维域（如条件均值）具有显着优势。最后，我们证明了 ML-EnCMF 在混乱状态下跟踪 Lorenz-63 和 Lorenz-96 系统状态的有效性。数值结果表明，ML-EnCMF 优于集成卡尔曼滤波器。