| Summary: | In this paper, we propose Deep Data Assimilation (DDA), an integration of Data Assimilation (DA) with Machine Learning (ML). DA is the Bayesian approximation of the true state of some physical system at a given time by combining time-distributed observations with a dynamic model in an optimal way. We use a ML model in order to learn the assimilation process. In particular, a recurrent neural network, trained with the state of the dynamical system and the results of the DA process, is applied for this purpose. At each iteration, we learn a function that accumulates the misfit between the results of the forecasting model and the results of the DA. Subsequently, we compose this function with the dynamic model. This resulting composition is a dynamic model that includes the features of the DA process and that can be used for future prediction without the necessity of the DA. In fact, we prove that the DDA approach implies a reduction of the model error, which decreases at each iteration; this is achieved thanks to the use of DA in the training process. DDA is very useful in that cases when observations are not available for some time steps and DA cannot be applied to reduce the model error. The effectiveness of this method is validated by examples and a sensitivity study. In this paper, the DDA technology is applied to two different applications: the Double integral mass dot system and the Lorenz system. However, the algorithm and numerical methods that are proposed in this work can be applied to other physical problem that involves other equations and/or state variables.
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