Combining data assimilation and machine learning to infer unresolved scale parametrisation

Julien Brajard 1,2, Alberto Carrassi 3,4, Marc Bocquet 5 and Laurent Bertino 1 1 Nansen Center (NERSC), 5006, Bergen, Norway 2 Sorbonne University, Paris, France 3 Department of Meteorology, University of Reading and NCEO, United-Kingdom 4 Mathematical Institute, University of Utrecht, The Netherlands 5 CEREA, joint laboratory École des Ponts ParisT ech and EDF R&D, Université Paris-Est, France In recent years, machine learning (ML) has been proposed to devise data-driven parametrisations of unresolved processes in dynamical numerical models. In most cases, the ML training leverages high-resolution simulations to provide a dense, noiseless target state. Our goal is to go beyond the use of high-resolution simulations and train MLbased parametrisation using direct data, in the realistic scenario of noisy and sparse observations. The algorithm proposed in this work is a two-step process. First, data assimilation (DA) techniques are applied to estimate the full state of the system from a truncated model. The unresolved part of the truncated model is viewed as a model error in the DA system. In a second step, ML is used to emulate the unresolved part, a predictor of model error given the state of the system. Finally, the MLbased parametrisation model is added to the physical core truncated model to produce a hybrid model. The DA component of the proposed method relies on an ensemble Kalman filter while the ML parametrisation is represented by a neural network. The approach is applied to the two-scale Lorenz model and to MAOOAM, a reduced-order coupled ocean-atmosphere model.