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Latent space data assimilation by using deep learning

This thesis, which sits at the crossroads of data assimilation (DA) and deep learning (DL), introduces latent space data assimilation, a novel data-driven framework that significantly reduces computational costs and memory requirements, while also offering the potential for more accurate data assimilation results. Importantly, this method is agnostic to the specific data assimilation algorithm and neural network architecture used.

We extend the integration of deep learning by rethinking the assimilation process itself. Our approach aligns with reduced-space methods, which solve the assimilation problem by performing computations within a lower-dimensional space relative. These reduced-space methods have been developed primarily for operational use, as most data assimilation algorithms are prohibitively computationally costly, when implemented in their full theoretically form.

Our methodology is based on the joint training of an autoencoder and a surrogate neural network. The autoencoder iteratively learns how to accurately represent the physical dynamics of interest within a low-dimensional space, termed latent space. The surrogate is simultaneously trained to learn the time propagation of the latent variables. A chained loss function strategy is also proposed to ensure the stability of the surrogate network. Stability can also be achieved by implementing Lipschitz surrogate networks.

Reduced-space data assimilation is underpinned by Lyapunov stability theory, which mathematically demonstrates that, under specific hypotheses, the forecast and posterior error covariance matrices asymptotically conform to the unstable-neutral subspace, which is of much smaller in dimension than the full state space. While full-space data assimilation involves linear combinations within a high-dimensional, nonlinear, and possibly multi-scale dynamic environment, latent data assimilation, which operates on the core, potentially disentangled and simplified dynamics, is more likely to result in impactful corrections. Furthermore, classical data assimilation faces an inherent mathematical limitation in that it relies on linear computations. Latent data assimilation, however, offers a way to overcome this limitation by performing the assimilation directly within the meaningful underlying structures of the data, obtained through nonlinear transformations.

We test our methodology on a 400-dimensional dynamics – built upon a chaotic Lorenz96 system of dimension 40 -, and on the quasi-geostrophic model of the Object-Oriented Prediction System (OOPS) framework, developed by the ECMWF (European Centre for Medium-Range Weather Forecasts).

Jury

Arthur VIDARD	Research Fellow at INRIA Grenoble	Reviewer
Tijana JANJIC	Professor at the Catholic University of Eichstätt-Ingolstadt	Reviewer
Ronan FABLET	Professor at IMT Atlantique	Examiner
Alban FARCHI	Researcher at ECMWF	Examiner
Selime GÜROL	Researcher at CERFACS	Supervisor
Serge GRATTON	University Professor at Toulouse INP	Co-Supervisor