# Deep Data Assimilation: Integrating Deep Learning with Data Assimilation
## 1 Introduction and Motivations
For this reason, a good quality of the data (smaller errors) can help to generate a more reliable DNN. 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. In some sense, DA increases the meaningfulness of the data and reduces the forecasting errors. Therefore, it is interesting to investigate the mechanism where two data driven paradigms, DA and DNN, can be integrated. In this context, we developed the Deep Data Assimilation (DDA) model. DDA is a new concept that combines the DNN and DA. It faces the problem to reduce both input data error (by DA) and modelling error (by ML). Related Works and Contribution of the Present WorkThis paper is placed in the imperfect models and sensitivity analysis context for data assimilation and deep neural network. Sensitivity Analysis (SA) refers to the determination of the contributions of individual uncertainty on data to the uncertainty in the solution [11]. The sensitivity of the variational DA models has been studied in [12], where an Adjoint modeling is used in order to obtain first and second-order derivative information. In [13], sensitivity analysis is based on the Backward Error Analysis (B.E.A. ), which figure out how much the errors propagation in DA process depend on the condition number of the background error covariance matrices. Reduced-order approaches are formulated in [12] in order to alleviate the computational cost that is associated with the sensitivity estimation. This method makes rerunning less expensive, the parameters must still be selected a priori and, consequently, important sensitivities may be missed [14]. For imperfect models, weak constraint DA (WCDA) can be employed [15], in which the model error term in the covariance evolution equation acts to reduce the dependence of the estimate on observations and prior states that are well separated in time. However, the time complexity of the WCDA models is a big limit that makes these models often unusable. ML algorithms are capable of assisting or replacing, without the assumptions of traditional methods, the aforementioned conventional methods in assimilating data, and producing forecasts. Because any linear or nonlinear functions can be approximated by neural networks, DA has been integrated as a supplement in various applications. The technology and the model presented in this paper are very general and they can be applied to any kind of dynamical systems. The use of ML in correcting model forecasts is promising in several geophysics applications [16; 17]. [PERSON] [18; 19; 20] applies neural network for error correction in forecasting. However, in this literature, the error correction neural network has not a direct relation with the system update model in each step and it does not train the results of the assimilation process. In [21], DA and ML are also integrated and a surrogate model is used in order to replace the dynamical system. In this paper we do not replace the the dynamical system with a surrogate models, as this would add approximation errors. In fact, in real case scenarios (i.e., in some operational centres), data driven models are welcome to support Computational Fluid Dynamics simulations [9; 10; 22], but the systems of Partial Differential Equations that are stably implemented to predict (weather, climate, ocean, air pollution, et al.) dynamics are not replaced due to the big approximations that this replacement would introduce. The technology that we propose in this paper integrates DA with ML in a way that can be used in real case scenarios for real world applications without changing the already implemented dynamical systems. Other works have already explored the relationship between the learning, analysis, and implementation of data in ML and DA. These studies have concentrated on fusing methods from the two fields instead of taking a modular approach like the one implemented in this paper, thereby developing methods that are unique to those approaches.