# The Euler-Type Universal Numerical Integrator (E-TUNI) with Backward Integration
## 2 Symbols and Notations Adopted
\(tan^{k}\dot{\alpha}^{i}_{j}=tan^{+}\dot{\alpha}^{i}_{j}=\frac{lim}{\Delta t_{ \Delta t}-\frac{1}{y^{i}_{j}-1}y^{i}_{j}}{\Delta t}\)--Scalar positive instantaneous derivative for \(j=1,2,\dots,n\) at instant \(t_{k}\). \(tan^{-}\dot{\alpha}^{i}=[tan^{-}\dot{\alpha}^{i}_{1}\quad tan^{-}\dot{ \alpha}^{i}_{2}\dots tan^{-}\dot{\alpha}^{i}_{n}]^{T}\)--Vector of negative instantaneous derivatives at instant \(t_{k}\). \(tan^{-}\dot{\alpha}^{i}_{j}=\frac{lim}{\Delta t_{\Delta t}-\frac{1}{y^{i}_{j}- 1}y^{i}_{j}}{\Delta t}\)--Scalar negative instantaneous derivative for \(j=1,2,\dots,n\) at time \(t_{k}\). \(t_{k}\)--Time instant \(t_{k}=t_{0}+k\cdot\Delta t\). \(t_{k+1}\)--Time instant \(t_{k+1}=t_{0}+(k+1)\cdot\Delta t\). \(\Delta t\)--Integration step. \(i\)--Over-index that enumerates a particular curve from the family of curves of the dynamical system to be modelled (\(i=1,2,\ldots,q\)). \(j\)--Under-index that enumerates the state and control variables. \(k\)--Over-index that enumerates the discrete time instants (\(k=1,2,\ldots,r\)). \(r\)--Total number of horizons of the time variable. \(q\)--Total number of curves from the family of curves curves of the dynamic system to be modelled. \(t_{k}^{*}\)--Instant of time within the interval \([t_{k},t_{k+1}]\) as a result of the Differential Mean Value Theorem (see Theorem 2). \(t_{k}^{*}\)--Instant of time within the interval \([t_{k},t_{k+1}]\) as a result of the Integral Mean Value Theorem (see Theorem 3). To understand this symbology, the first step is to understand the difference between \({}^{k+1}y^{j}\), \({}^{k+1}\dot{y}^{i}\), and \({}^{k+1}\dot{y}^{i}\). The vector variable \({}^{k+1}\dot{y}^{i}\) is the exact value of the state variables at time \(t_{k+1}=t_{0}+(k+1)\cdot\Delta t\). The vector variable \({}^{k+1}\dot{y}^{i}\) is the estimated value of the state variables using only the numerical integrator. Finally, the vector variable \({}^{k+1}\dot{y}^{i}\) is the estimated value of the state variables using the numerical integrator coupled with an artificial neural network. Figures 1-4 follow this convention for a better understanding of the model proposed in this work. Additionally, it is important to understand the meaning of the auxiliary variables \(i\), \(j\), and \(k\). This is described in the next paragraph. Here, an important issue is to elucidate the use of the over-indexes \(k\) and \(i\) and the sub-index \(j\) in the variables \({}^{k}y^{i}_{j}\) and \(tan_{\Delta t}{}^{k}a^{i}_{j}\). Figure 1 will help in this explanation. When the auxiliary variables \(i\), \(j\), and \(k\) are used simultaneously, they uniquely identify the secant \((tan_{\Delta t}{}^{k}a^{i}_{j})\) at the point \({}^{k}\dot{y}^{i}_{j}\). Notice that the over-index \(k\) indicates the time instant \(t_{k}=t_{0}+k\cdot\Delta t\) of the secant, the sub-index \(j\) indicates the state variable in question, where \(j=1,2,\ldots,n\), and the over-index \(i\) (\(i=1,2,\ldots,q\)) indicates the particular curve, where the respective secant is located, of the family of possible curves of the system of differential equations considered. The value of \(q\) can be as large as one wants. The larger the value of \(q\) the more different curves will be trained by the neural network, and the better its generalization will be. This unique mapping of the secant \(tan_{\Delta t}{}^{k}a^{i}_{j}\) will be of fundamental importance for the demonstration of Theorem 4 that will be presented and demonstrated in Section 5. This theorem establishes that if the Euler integrator uses the secant instead of the instantaneous derivatives, it will be a discrete and exact solution for a system of ordinary and autonomous differential equations. Furthermore, the mapping mentioned above is unique because of  
Figure 1: Symbology and notation used in this article. the uniqueness theorem for systems of ordinary differential equations. Finally, an artificial neural network will be used to learn the secants of the considered dynamic system through supervised learning using input/output training patterns.