# Deep Data Assimilation: Integrating Deep Learning with Data Assimilation
## 3 The Deep Data Assimilation (DDA) Model
We assume  
\[\|B_{k}\|_{\infty}>\|B_{k-1}\|_{\infty} \tag{21}\]  
At step \(k-1\), from Step 10 of Algorithm 2 we have \(B_{k}=A_{k-1}\), i.e., from (15) and (13), the (21) gives  
\[\left\|\left(I-\frac{H_{k-1}B_{k-1}H_{k-1}^{T}}{H_{k-1}B_{k-1}H_{k-1}^{T}+R_{k- 1}}\right)B_{k-1}\right\|_{\infty}>\|B_{k-1}\|_{\infty} \tag{22}\]  
i.e., from the property of \(\|\cdot\|_{\infty}\): \(\|a\|_{\infty}\|b\|_{\infty}>\|a\ b\|_{\infty}\), we have  
\[\left\|I-\frac{H_{k-1}B_{k-1}H_{k-1}^{T}}{H_{k-1}B_{k-1}H_{k-1}^{T}+R_{k-1}} \right\|B_{k-1}\|_{\infty}>\|B_{k-1}\|_{\infty} \tag{23}\]  
which means that  
\[\left\|I-\frac{H_{k-1}B_{k-1}H_{k-1}^{T}}{H_{k-1}B_{k-1}H_{k-1}^{T}+R_{k-1}} \right\|_{\infty}>1 \tag{24}\]  
Figure 5: Recurrent Neural Network. Because of the definition of \(\|\cdot\|_{\infty}\), we can then assume that \(\exists\ ij\), such that  
\[1-\frac{b_{ij}}{b_{ij}+r_{ij}}>1 \tag{25}\]  
i.e., such that  
\[\frac{b_{ij}}{b_{ij}+r_{ij}}<0. \tag{26}\]  
which is absurd. In fact, if we consider the two possible conditions \(i=j\) and \(i\
eq j\). In case \(i=j\), \(b_{ii}\) and \(r_{ii}\) are diagonal elements of the errors covariance matrices. In other words, they are values of variances that mean that they are positive values and the (26) is not satisfied. In case \(i\
eq j\), \(r_{ij}=0\) as the error covariance matrix \(R_{k}\) is diagonal, as defined in (8). Subsequently, we have that \(\frac{b_{ij}}{b_{ij}+r_{ij}}=\frac{b_{ij}}{b_{ij}}=1\) and the (26) is also not satisfied. Subsequently, the (20) is proven. Another important aspect is that the solution of the DDA Algorithm 2 is more accurate than the solution of a standard DA Algorithm 1. In fact, even if some sources of errors cannot be ignored (i.e., the round off error), then the introduction of the DDA method reduces the error in the numerical forecasting model (below denoted as \(\xi_{k}\)). The following result held. Theorem 2: _Let \(\delta_{k}\) denote the error in the DA solution:_  
\[u_{k}^{DA}=u_{k}^{true}+\delta_{k}\;,\]  
_and let \(\delta_{k}\) be the error in DDA solution:_  
\[\|B_{k}\|_{\infty}\leq\|B_{k-1}\|_{\infty} \tag{20}\]0  
_it is:_  
\[\|B_{k}\|_{\infty}\leq\|B_{k-1}\|_{\infty} \tag{20}\]1  
Proof. : Let \(\zeta_{k}\) be error in the solution of the dynamic system \(M\) in (3), i.e., \(u_{k}=M(u_{k-1})+\xi_{k}\). From the backward error analysis that was applied to the DA algorithm [13], we have:  
\[\|B_{k}\|_{\infty}\leq\|B_{k-1}\|_{\infty} \tag{20}\]2  
and  
\[\|B_{k}\|_{\infty}\leq\|B_{k-1}\|_{\infty} \tag{20}\]3  
where \(\mu_{DA}\) and \(\mu_{DDA}\) denote the condition numbers of the DA numerical model and the DDA numerical model, respectively. It has been proved, in [37], that the condition number of the DA and DDA numerical models are directly proportional to the condition numbers of the related background error covariance matrices \(B_{k}\). Subsequently, thanks to the (20), we have \(\mu_{DDA}\leq\mu_{DA}\), from which:  
\[\|B_{k}\|_{\infty}\leq\|B_{k-1}\|_{\infty} \tag{20}\]4  
The DDA framework in this article implement a Recurrent neural network (RNN) structure as \(\mathcal{G}_{\omega}\). The loss function is defined, as described in Theorem 3 and the network is updated at each step by the gradient loss function on its weights and parameters:  
\[\|B_{k}\|_{\infty}\leq\|B_{k-1}\|_{\infty} \tag{20}\]5  
**Theorem 3**. : _Let \(\{(u_{k},u_{k}^{PDA})\}\) be the training set for \(\mathcal{G}_{\omega,i}\), which exploits the data \(\{(u_{k-1}^{PDA},u_{k}^{PDA})\}\) from the past updated model \(\mathcal{M}_{i}\) in (17). The loss function for the Back-propagation, being defined as mean square error (MSE) for the \(DA\) training set, is such that_  
\[\|B_{k}\|_{\infty}\leq\|B_{k-1}\|_{\infty} \tag{20}\]6  
_where \(d_{k}=v_{k}-H_{k}u_{k}\) is the misfit between observed data and background data, and \(O_{k}=H_{k}B_{k}H_{k}^{T}+R_{k}\) is the Hessian of the \(DA\) system._  
Proof.