# Deep Data Assimilation: Integrating Deep Learning with Data Assimilation
## 4 Experiments  
In this section, we apply the DDA technology to two different applications: the Double integral mass dot system and the Lorenz system:  
* Double integral mass dot system: For the physical representation of a generalized motion system, the double-integral particle system is commonly used. The method, under the influence of an acceleration regulation quantity, can be seen as the motion of a particle. The position and velocity are the variables that affect the state at each time step. The status as a controlled state gradually converges to a relatively stable value, due to the presence of a PID controller in the feedback control system. The performance of the controller is the system-acting acceleration, or it can be interpreted as a force that linearly relates to the acceleration. Double integral system is a mathematic abstraction of [PERSON]'s Second Law that is often used as a simplified model of several controlled systems. It can be represented as a continuous model, as: \[\begin{split}\dot{u}&=Au+Bz+\xi,\\ v&=Cu+r\end{split}\] (36) where the state \(u=[u_{1},u_{2}]^{T}\) is a two-dimensional vector containing position \(u_{1}\) and velocity \(u_{2}\) and where \(z=\dot{u}_{2}\) is the controlled input. The coefficients matrices \(A\), \(B\), and \(C\) are time-invariant system and observations matrices. \(r\) is the noise of the observations, which is Gaussian and two dimensional. The system disruption \(\xi\) comprises two aspects: random system noise and instability of the structural model.
* Lorenz system: [PERSON] developed a simplified mathematical model for atmospheric convection in 1963. It is a common test case for DA algorithms. The model is a system of three ordinary differential equations, named Lorenz equations. For some parameter values and initial conditions, it is noteworthy for having a chaotic behavior. The Lorenz equations are given by the nonlinear system:\[\frac{dp}{dt} =-\sigma(p-q),\] \[\frac{dq}{dt} =\rho p-q-pr, \tag{37}\] \[\frac{dr}{dt} =pq-\beta r.\]  
where \(p\), \(q\) and \(r\) are coordinates, and \(\sigma\), \(\rho\), and \(\beta\) are parameters. In our implementation, the model has been discretized by second order Runge-Kutta method [43] and, for this test case, we posed \(\sigma=10\), \(\rho=8/3\) and \(\beta=28\).  
We mainly provide testing in order to validate the results that we proved in the previous section. Afterwards, we mainly focus on:  
* DDA accuracy: we prove that the accuracy of the forecasting result increases as the number of time steps increases, as proved in Theorem 1;
* DDA vs. DA: in order to address the true models, we face the circumstances where DA is not sufficient to reduce the modeling uncertainty and we show that the DDA algorithm, we have proposed is applicable to solve this issue. Such findings validate what we showed in Theorem 2; and,
* R-DDA vs. F-DDA: we are demonstrating that the accuracy of the R-DDA is better than that of the F-DDA. It is also due to the way the weight is calculated, i.e., by including the DA covariance matrices, as shown in Theorem 3.  
We implemented the algorithms on the Simulink (Simulation and Model-Based Design) in Matlab 2016a.