# The Euler-Type Universal Numerical Integrator (E-TUNI) with Backward Integration
## 3 Contextualization  
This section clarifies some critical issues regarding using the E-TUNI in mathematics, physics, and engineering problems. These issues will help provide a better theoretical and practical understanding of the method proposed in this article. To this end, in the following, we formulate five essential questions regarding the proper design of E-TUNI and give succinct and objective answers to all of them. What is the difference between E-TUNI and the Conventional Euler method? The traditional Euler method works with instantaneous derivative functions, and the E-TUNI works with mean derivative functions. Although this difference may seem insignificant, in reality, it is not. We also can refer to Figure 2 to answer this question. In this figure, \({}^{k+1}\tilde{y}^{i}_{j}\), \({}^{k+1}\tilde{y}^{i}_{j}\) and \({}^{k+1}y^{i}_{j}\) are, respectively, the solution obtained by conventional Euler, the solution obtained by E-TUNI, and the exact solution of the dynamical system at time \(t_{k+1}\). As can be seen, the mean derivative gives a more accurate solution than the instantaneous derivative. Another advantage of E-TUNI is that it can be designed using real-world training patterns through supervised learning. To this end, E-TUNI uses an artificial neural network to learn the mean derivative functions with high precision. Notice also a difference between the values \({}^{k+1}\tilde{y}^{i}_{j}\) and \({}^{k+1}y^{i}\). This difference is caused by the training of the neural network since the error generated by this same neural network is non-zero. What are the advantages of E-TUNI with backward integration compared to other conventional numerical integration methods? Since E-TUNI uses a neural network, its accuracy is equivalent to any other higher-order numerical integrator. This property is because artificial neural networks are universal approximators of functions [2; 3; 4; 5]; that is, the training error of the neural network can be as small as desired. The other advantage of E-TUNI is that it has a mathematical structure only of the first-order, which means it is much simpler than higher-order integrators. The disadvantage of E-TUNI, concerning higher-order integrators, is that the former has a fixed step, and the latter has a variable step. This difference means that if it is desired to vary the integration step of E-TUNI, a new neural training will be necessary. Furthermore, as previously mentioned, E-TUNI can work with real-world data and not only with theoretical models. How does the algorithm guarantee accuracy and stability when finding the endpoints of the integration interval during the backward integration process? In this paper, we only empirically verify that this is possible through the practical examples presented in the  
Figure 4: The E-TUNI designed according to the direct approach (Source: [16; 27]). numerical results section at the end of this paper. However, a convergence proof of the numerical stability of the proposed model is left for future work. However, here, we prove that the general expression of E-TUNI (which works with mean derivative functions) is an exact solution for a system of Ordinary Differential Equations (ODEs) and as demonstrated by Theorem 4. Is this valuable method for dynamical systems of nonlinear differential equations of any dimension \(n\)? As exemplified by Equations (1)-(3), E-TUNI is capable of solving a non-linear dynamic modelling problem of any order. However, it seems reasonable to assume that the larger the dimension \(n\) of the proposed problem, the greater the computational effort required to train the neural network used. This article exemplifies its technological application with physics problems of up to four state variables. Does the E-TUNI model have application value for new problems in a specific discipline such as Physics or Engineering? Yes, this method can be applied to any physical problem governed by an autonomous non-linear ordinary differential equation with \(n\) state variables and \(m\) control variables.