# The Euler-Type Universal Numerical Integrator (E-TUNI) with Backward Integration
## 5 The Euler-Type Universal Numerical Integrator (E-TUNI)
### Fundamental Theorem  
In this section, the general expression of E-TUNI is demonstrated. For this purpose, let the system of non-linear differential Equations (5) be:  
\[\dot{y}=f(y) \tag{5}\]  
where, \(y=[y_{1}\quad y_{2}\quad\dots\quad y_{n}]^{T}\) and \(f(y)=[f_{1}(y)\quad f_{2}(y)\quad\dots\quad f_{n}(y)]^{T}\). The uniqueness theorem is a fundamental theorem about Equation (5). Theorem 1 (T1): _Assume that each of the functions \(f_{1}(y_{1},y_{2},\dots,y_{n}),\dots,f_{n}(y_{1},y_{2},\dots,y_{n})\) has continuous partial derivatives with respect to \(y_{1},y_{2},\dots,y_{n}\). So the known initial value problem \(\dot{y}=f(y)\) for \(y(t_{0})\) has one and only one solution \(y=y(t)\) in \(\mathbb{R}\), starting from each \(y(t_{0})\). If two solutions \(y=\phi(t)\) have a common point, they must be identical._  
For the formal demonstration of the general expression of E-TUNI, which will numerically and discretely solve Equation (5), two hypotheses will be assumed: (i) a set of points that is an exact solution of the dynamical system (5) is known and which we will call \({}^{k}y_{j}^{i}\) and (ii) the existence of two sets of continuous and differentiable functions that pass through these points is assumed, namely, \(y_{j}^{i}(t)\) and \(y_{j}^{i}(t)\). Figure 7 mathematically illustrates these two hypotheses when \(i=1,2,3,4\), \(j=1,2\), and \(k=0,1,2,3,4\). The meanings of the auxiliary variables \(i\), \(j\), and \(k\) are the same as those described in Figure 1. Furthermore, only the points \({}^{k}y_{j}^{i}\) must be known. The functions \(y_{j}^{i}(t)\) and \(y_{j}^{i}(t)\) just need to exist; that is, it is not necessary to know them. What do the matrices mean in Figure 7? The first line of these point matrices represents a set of exact points belonging to the first curve of the family of solution curves of the considered system, and so on. For example, all points \({}^{0}y_{1}^{1}\), \({}^{1}y_{1}^{1}\), \({}^{2}y_{1}^{1}\), \({}^{3}y_{1}^{1}\), and \({}^{4}y_{1}^{1}\) pass through the continuous and differentiable function \(y_{1}^{1}(t)\), and so on. Since \(i=4\), the family of curves, in this case, is composed of four curves. Based on the continuous hypothesis, \(i\) can be as large as one wants. Since the neural network, responsible for modelling the system (5), will be designed with the supervised learning approach through input/output training patterns, knowing in advance the discrete solution of the system (5), is not absurd. Thus, the system (5) can be replaced by the following set of solutions (6):  
\[ \begin{cases}\dot{y}_{j}^{i}(t)=\dot{g}_{j}^{i}(t)\\ \dot{y}_{j}^{i}(t)=\dot{g}_{j}^{i}(t)\end{cases}  \tag{6}\]  
where \(i=1,2,\dots,q\); \(j=1,2,\dots,n\); \(k=0,1,2,\dots,r\). It is essential to note that changing the values of \(q\), \(n\), \(r\), and \(\Delta t\) can obtain any finite density of discrete point solutions of (5) over a finite domain. Thus, the great advantage of Equation (6) concerning Equation (5) is that the former is an explicit function only of the variable \(t\). Thus, the differential and integral mean  
Figure 7: Example of numerical discretization of the solution of the differential Equation (5). value theorems can be applied in (6) to arrive at the general expression of E-TUNI, which uses the functions of mean derivatives. The differential and integral mean value theorems cannot be applied in (5) because, in this case, the system in question is an explicit function of \(y\) and not of \(t\). **Theorem 2** (The Differential Mean Value Theorem). : _If a function \(g^{i}_{j}(t)\) for \(j=1,2,\ldots,n\) is a continuous function and defined over the closed interval \([t_{k},t_{k+1}]\) is differentiable over the open interval \((t_{k},t_{k+1})\), then there exists at least one number \(t_{k}^{*}\) with \(t_{k}<t_{k}^{*}<t_{k}+\Delta t=t_{k+1}\) such that, \(\delta^{i}_{j}(t_{k}^{*})=\frac{{}^{k+1}\delta^{i}_{j}-{}^{k}\delta^{i}_{j}}{ \Delta t}\)._  
**Theorem 3** (The Integral Mean Value Theorem).