# Deep Data Assimilation: Integrating Deep Learning with Data Assimilation
## 1 Introduction and Motivations
The integration of the DA and ML frameworks from a Bayesian perspective illustrates the interface between probabilistic ML methods and differential equations. The equivalence is formally presented in [23] and it illustrates the parallels between the two fields. Here, the equivalence of four-dimensional VarDA (4D-Var) and Recurrent NN, and how approximate Bayesian inverse methods (i.e., VarDA in DA and back-propagation in ML) can be used to merge the two fields. These methods are also especially suited to systems involving Gaussian process, as presented in [24; 25; 26]. These are developed data-driven algorithms that are capable, under a unified approach, of learning nonlinear, space-dependent cross-correlations, and of estimating model statistics. A DA method can be used via the DL framework to help the dynamic model (in this case, the ML model) to find optimal initial conditions. Although DA can be computationally costly, the recent dimensional reduction developments using ML substantially reduce this expense, while retaining much of the variance of the original [27] model. Therefore, DA helps to develop the ML model, while [28; 29] also benefits from the ML techniques. Methods that merge DA models, such as Kalman filters and ML, exist to overcome noise or to integrate time series knowledge. The authors account for temporal details in human motion analysis in [30] by incorporating a Kalman filter into a neural network of LSTM. In [31], the authors suggest a methodology that combines NN and DA for model error correction. Instead, in [32], the authors use DA, in particular Kalman tracking, to speed up any learning-based motion tracking method to real-time and to correct some common inconsistencies in motion tracking methods that are based on the camera. Finally, in [33], the authors introduce a new neural network for speed and replace the whole DA process. In this paper, we proposed a new concept, called DDA, which combines the DL into the conventional DA. In recent years, DL gained great success both in academic and industrial areas. In function approximation, which has unknown model and high nonlinearity, it shows great benefit. Here, we use DNN to describe the model uncertainty and the assimilation process. In an end-to-end approach, the neural networks are introduced and their parameters are iteratively modified by applying the DA methods with upcoming observations. The resulting DNN model includes the features of the DA process. We prove that this implies a reduction of the model error, which decreases at each iteration. We also prove that the DDA approach introduces an improvement in both DA and DNN. We introduce a sensitivity analysis that is based on the backward error analysis to compare the error in DDA result and the error in DA. We prove that the error in the results obtained by DDA is reduced with respect the DA. We also prove that the use of the results of DA to train the DL model introduces a novelty in the SA techniques used to evaluate which inputs are important in NN. The implemented approach can be compared to the 'Weights' Method [34]. The distinction from the classical 'Weights' method is that the weights between the nodes are given in our approach by the error covariance matrices that are determined in the DA phase. In summary, we prove that the DDA approach  
* includes the features of the DA process into the DNN model with a consequent reduction of the model error;
* introduces a reduction in the overall error in the assimilation solution with respect to the DA solution; and,
* increases the reliability of the DNN model, including the error covariance matrices as weight in the loss function. This paper is structured, as follows. Section 2 provides preliminaries on DA. We proposed the DDA methodology in Section 3, where we introduce the integration of deep learning with data assimilation. In Section 4, numerical examples are provided in order to implement and verity the DDA algorithm and, in Section 5, conclusions and future work are summarized.