# The Euler-Type Universal Numerical Integrator (E-TUNI) with Backward Integration
## 7 Results and Analysis  
In this section, we perform a numerical and computational analysis of the Universal Numerical Integrator (UNI), which exclusively uses mean derivative functions. As previously described, this is the Euler-Type Universal Numerical Integrator (E-TUNI). Therefore,we present here three case studies: (i) a one-dimensional simple case, (ii) the non-linear simple pendulum (two state variables), and (iii) the non-linear inverted pendulum (four state variables). Thus, for all experiments that it is presented below, the use of MLP neural networks was standardized with the traditional Levenberg-Marquardt training algorithm [33] from 1994. Concerning the architecture of the neural network used, all experiments were trained using only one inner layer (with thirty neurons with tangent hyperbolic activation function) in an artificial neural network of the Multi-Layer Perceptron (MLP) type. In all experiments, 80% of the total patterns were left for training the MLP neural network, 10% of the patterns for testing, and 10% for validation. Concerning the architecture of E-TUNI, the direct approach was exclusively used. However, it must be said that the indirect or empirical approach of E-TUNI was not tested in this article. Thus, the general algorithm for training and testing the methodology presented in this article can be briefly described as follows:  
Step 1. Using a high-order 4-5 Runge-Kutta integrator, over the considered autonomous non-linear ordinary differential equation system, randomly generate \(p\) training patterns input/output for E-TUNI concerning the initial instant \(y^{i}(t_{0})\). Perform both forward and backward integration. Step 2. Determine both the values of the positive mean derivatives \(tan_{\Delta t}^{+}\kappa_{j^{\prime}}^{i}\) as well as the values of the negative mean derivatives \(tan_{\Delta t}^{-}\kappa_{j}^{i}\) in relation to initial instants \(y^{i}(t_{0})\) for \(i=1,2,\ldots,p\). Step 3. Train two distinct neural networks. One is to output positive mean derivative functions, and the other is to output negative mean derivative functions. Do this following the graphic diagram in Figure 6.  
Step 4. Simulate, for several different \(\Delta t\) integration steps, the performance of E-TUNI both with forward and backward integration. Observe that if starting from an initial instant \(y^{i}(t_{0})\), the solution is propagated in \(n\) distinct instants forward and, after that, one returns to the same \(n\) instants with backward integration because of the uniqueness theorem, one returns perfectly to the original initial instant \(y^{i}(t_{0})\). This trick will be used to test the efficiency of E-TUNI when it is propagated backward. Step 5. Do the same, using the Euler-type first-order integrator, but now using the original instantaneous derivative functions of the dynamical system in question. Then, compare E-TUNI with conventional Euler. Thus, to verify the superiority of E-TUNI concerning the conventional Euler, the accuracy of the discrete numerical solutions obtained must be verified. Therefore, next, we briefly describe the systems of non-linear differential equations of the three dynamical systems considered here to test the proposed algorithm to perform the backward integration of E-TUNI. Table 1 also summarizes the training data of the eight experiments conducted here to test the proposed methodology. **Example 1**. : _Simulate the one-dimensional non-linear dynamics problem, given by Equation (35). The variable \(y\) was trained on the range \([-1,13]\). Note that this dynamic has infinite stability points since \(siny=0\) has infinitely many solutions. Note that the points of stability recovering the initial condition through backward integration is very difficult for any integrator._  
\[\dot{y}=sin(y) \tag{35}\]  
**Example 2**. : _Simulate the non-linear simple pendulum problem given by the system of differential Equations (36). Note that this example does not involve control variables.