Data assimilation techniques are crucial for correcting the trajectory when modeling complex physical systems. A recently developed data assimilation method, Latent Ensemble Score Filter (Latent-EnSF), has shown great promise in addressing the key limitation of EnSF for highly sparse observations in high-dimensional and nonlinear data assimilation problems. It performs data assimilation in a latent space for encoded states and observations in every assimilation step, and requires costly full dynamics to be evolved in the original space. In this paper, we introduce Latent Dynamics EnSF (LD-EnSF), a novel methodology that completely avoids the full dynamics evolution and significantly accelerates the data assimilation process, which is especially valuable for complex dynamical problems that require fast data assimilation in real time. To accomplish this, we introduce a novel variant of Latent Dynamics Networks (LDNets) to effectively capture and preserve the system's dynamics within a very low-dimensional latent space. Additionally, we propose a new method for encoding sparse observations into the latent space using Long Short-Term Memory (LSTM) networks, which leverage not only the current step's observations, as in Latent-EnSF, but also all previous steps, thereby improving the accuracy and robustness of the observation encoding. We demonstrate the robustness, accuracy, and efficiency of the proposed method for two challenging dynamical systems with highly sparse (in both space and time) and noisy observations.

Accurate modeling and prediction of complex physical systems often rely on data assimilation techniques to correct errors inherent in model simulations. Traditional methods like the Ensemble Kalman Filter (EnKF) and its variants as well as the recently developed Ensemble Score Filters (EnSF) face significant challenges when dealing with high-dimensional and nonlinear Bayesian filtering problems with sparse observations, which are ubiquitous in real-world applications. In this paper, we propose a novel data assimilation method, Latent-EnSF, which leverages EnSF with efficient and consistent latent representations of the full states and sparse observations to address the joint challenges of high dimensionlity in states and high sparsity in observations for nonlinear Bayesian filtering. We introduce a coupled Variational Autoencoder (VAE) with two encoders to encode the full states and sparse observations in a consistent way guaranteed by a latent distribution matching and regularization as well as a consistent state reconstruction. With comparison to several methods, we demonstrate the higher accuracy, faster convergence, and higher efficiency of Latent-EnSF for two challenging applications with complex models in shallow water wave propagation and medium-range weather forecasting, for highly sparse observations in both space and time. Many complex physical systems are traditionally modeled by partial differential equations (PDEs).