Pierre Boudier

Abstract Data Assimilation aims at estimating the posterior conditional probability density functions based on error statistics of the noisy observations and the dynamical system. State of the art methods are sub‐optimal due to the common use of Gaussian error statistics and the linearization of the non‐linear dynamics. To achieve a good performance, these methods often require case‐by‐case fine‐tuning by using explicit regularization techniques such as inflation and localization. In this paper, we propose a fully data driven deep learning framework generalizing recurrent Elman networks and data assimilation algorithms. Our approach approximates a sequence of prior and posterior densities conditioned on noisy observations using a log‐likelihood cost function . By construction our approach can then be used for general nonlinear dynamics and non‐Gaussian densities. As a first step, we evaluate the performance of the proposed approach by using fully and partially observed Lorenz‐95 system in which the outputs of the recurrent network are fitted to Gaussian densities. We numerically show that our approach, without using any explicit regularization technique , achieves comparable performance to the state‐of‐the‐art methods, IEnKF‐Q and LETKF, across various ensemble size.

Abstract Performing data assimilation (DA) at low cost is of prime concern in Earth system modeling, particularly in the era of Big Data, where huge quantities of observations are available. Capitalizing on the ability of neural network techniques to approximate the solution of partial differential equations (PDEs), we incorporate deep learning (DL) methods into a DA framework. More precisely, we exploit the latent structure provided by autoencoders (AEs) to design an ensemble transform Kalman filter with model error (ETKF‐Q) in the latent space. Model dynamics are also propagated within the latent space via a surrogate neural network. This novel ETKF‐Q‐Latent (ETKF‐Q‐L) algorithm is tested on a tailored instructional version of Lorenz 96 equations, named the augmented Lorenz 96 system , which possesses a latent structure that accurately represents the observed dynamics. Numerical experiments based on this particular system evidence that the ETKF‐Q‐L approach both reduces the computational cost and provides better accuracy than state‐of‐the‐art algorithms such as the ETKF‐Q.

Data assimilation algorithms aim at forecasting the state of a dynamical system by combining a mathematical representation of the system with noisy observations thereof. We propose a fully data driven deep learning architecture generalizing recurrent Elman networks and data assimilation algorithms which provably reaches the same prediction goals as the latter. On numerical experiments based on the well-known Lorenz system and when suitably trained using snapshots of the system trajectory (i.e. batches of state trajectories) and observations, our architecture successfully reconstructs both the analysis and the propagation of probability density functions of the system state at a given time conditioned to past observations.