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.

Training normalizing flow generative models can be challenging due to the need to calculate computationally expensive determinants of Jacobians. This paper studies the likelihood-free training of flows and proposes the energy objective, an alternative sample-based loss based on proper scoring rules. The energy objective is determinant-free and supports flexible model architectures that are not easily compatible with maximum likelihood training, including semi-autoregressive energy flows, a novel model family that interpolates between fully autoregressive and non-autoregressive models. Energy flows feature competitive sample quality, posterior inference, and generation speed relative to likelihood-based flows; this performance is decorrelated from the quality of log-likelihood estimates, which are generally very poor. Our findings question the use of maximum likelihood as an objective or a metric, and contribute to a scientific study of its role in generative modeling. Code is available at https://github.com/ps789/SAEF.