Data assimilation is a key component of operational systems and scientific studies for the understanding, modeling, forecasting and reconstruction of earth systems informed by observation data. Here, we investigate how physics-informed deep learning may provide new means to revisit data assimilation problems. We develop a so-called end-to-end learning approach, which explicitly relies on a variational data assimilation formulation. Using automatic differentiation embedded in deep learning framework, the key novelty of the proposed physics-informed approach is to allow the joint training of the representation of the dynamical process of interest as well as of the solver of the data assimilation problem. We may perform this joint training using both supervised and unsupervised strategies. Our numerical experiments on Lorenz-63 and Lorenz-96 systems report significant gain w.r.t. a classic gradient-based minimization of the variational cost both in terms of reconstruction performance and optimization complexity. Intriguingly, we also show that the variational models issued from the true Lorenz-63 and Lorenz-96 ODE representations may not lead to the best reconstruction performance. We believe these results may open new research avenues for the specification of assimilation models for earth systems, both to speed-up the inversion problem with trainable solvers but possibly more importantly in the way data assimilation systems are designed, for instance regarding the representation of geophysical dynamics.

Plain Language Summary

Data assimilation is a key component in the modeling of earth systems to simulate their dynamics, forecast their evolution in the short-term or the long-term as well as to reconstruct earth systems' states from observation data. State-of-the-art data assimilation schemes generally blend prior knowledge on the underlying governing laws with available observation data. Here, we turn data assimilation into a physics-informed machine learning problem. Within a differentiable framework, we can learn from data not only a data assimilation solver but also jointly some representation of the inverse problem. Numerical experiments support the relevance of this end-to-end approach for chaotic dynamics informed by noisy and irregularly-sampled observations. This opens new research avenues for the design of physics-informed and data-constrained simulation, forecasting and reconstruction schemes for earth systems.