# The Euler-Type Universal Numerical Integrator (E-TUNI) with Backward Integration
## 2 Symbols and Notations Adopted  
For a more detailed understanding of the theoretical development presented in this paper, we define all the symbols and variables used in this work. The proposed method is called Euler-Type Universal Numerical Integrator (E-TUNI) and uses backward integration. This method is discrete. Thus, we propose a discrete method to approximate a continuous system of autonomous ordinary differential equations. Therefore, below, we present all the symbolologies and variables used here. They are divided into two groups: (i) continuous variables and (ii) discrete variables. **(i) Continuous Variables**  
\(\dot{y}=f(y)\)--System of continuous differential equations. \(y=[y_{1}\quad y_{2}\dots y_{n}]^{T}\)--State Variables. \(f(y)=[f_{1}(y)\quad f_{2}(y)\dots f_{n}(y)]^{T}\)--Instantaneous derivative functions. \(d\)--Plant noise. \(d^{\prime}\)--Measuring instrument noise. \(y^{i}_{j}(t)=g^{i}_{j}(t)\)--Particular continuous and differentiable curve of a family of solution curves of the dynamical system \(\dot{y}=f(y)\). \(\dot{y}^{i}_{j}(t)=\dot{g}^{i}_{j}(t)\)--First derivative of \(y^{i}_{j}(t)\). **(ii) Discrete Variables**  
\({}^{k}y^{i}=y^{i}(t_{0}+k\cdot\Delta t)\)--Vector of state variables at time \(t_{k}\). \({}^{k}y^{i}_{j}\)--Scalar state variable for \(j=1,2,\dots,n\) at time \(t_{k}\). It is a generic discretization point of the state variables generated by the integers \(i,j\), and \(k\). \(n\)--Total number of state variables. \({}^{k}u=u(t_{0}+k\cdot\Delta t)\)--Vector of control variables at time \(t_{k}\). \({}^{k}u_{j}\)--Scalar control variable for \(j=1,2,\dots,m\) at time \(t_{k}\). \(m\)--Total number of control variables. \({}^{k+1}y^{i}=y^{i}[t_{0}+(k+1)\cdot\Delta t]\)--Exact vector of state variables at time \(t_{k+1}\). \({}^{k+1}y^{i}_{j}\)--Exact scalar state variable for \(j=1,2,\dots,n\) at time \(t_{k+1}\). \({}^{k+1}\dot{g}^{i}=\dot{g}^{i}[t_{0}+(k+1)\cdot\Delta t]\)--Estimated Vector of state variables by UNI or E-TUNI at time \(t_{k+1}\). \({}^{k+1}\dot{g}^{i}_{j}\)--Scalar state variable estimated by UNI or E-TUNI for \(j=1,2,\dots,n\) at time \(t_{k+1}\). \({}^{k+1}\dot{g}^{i}\)--Estimated Vector of state variables when using only the integrator and without using the neural network at time \(t_{k+1}\). \(tan_{\Delta t}{}^{k}\alpha^{i}=tan_{\Delta t}^{+}\alpha^{i}=[tan_{\Delta t}^{ +}\alpha^{i}_{1}\quad tan_{\Delta t}^{+}\alpha^{i}_{2}\dots tan_{\Delta t}^{ +}\alpha^{i}_{n}]^{T}\)--Exact vector of positive mean derivative functions at time \(t_{k}\). \(tan_{\Delta t}{}^{k}\alpha^{j}_{j}=tan_{\Delta t}^{+}\alpha^{i}_{j}=\frac{- \frac{1}{y^{i}_{j}-y^{i}_{j}}}{\Delta t}\)--Scalar positive mean derivative functions for \(j=1,2,\dots,n\) at time \(t_{k}\). \(tan_{\Delta t}{}^{k}\dot{\alpha}^{i}=tan_{\Delta t}^{+}\dot{\alpha}^{i}=[tan_ {\Delta t}^{+}\dot{\alpha}^{i}_{1}\quad tan_{\Delta t}^{+}\dot{\alpha}^{i}_{2 }\dots tan_{\Delta t}^{+}\dot{\alpha}^{i}_{n}]^{T}\)--Estimated vector of positive mean derivative functions by the E-TUNI at time \(t_{k}\). \(tan_{\Delta t}{}^{k}\dot{\alpha}^{i}=[tan_{\Delta t}^{-}\dot{\alpha}^{k}_{1} \quad tan_{\Delta t}^{-}\dot{\alpha}^{k}_{2}\dots tan_{\Delta t}^{-}\dot{ \alpha}^{i}_{n}]^{T}\)--Exact vector of negative mean derivative functions at time \(t_{k}\). \(tan_{\Delta t}^{-}\dot{\alpha}^{i}_{j}=\frac{-\frac{1}{y^{i}_{j}-1}y^{i}_{j}}{ \Delta t}\)--Scalar negative mean derivative functions for \(j=1,2,\dots,n\) at time \(t_{k}\). \(tan_{\Delta t}{}^{-}\dot{\alpha}^{k}\dot{\alpha}^{i}=[tan_{\Delta t}^{-}\dot{ \alpha}^{k}_{1}\quad tan_{\Delta t}^{-}\dot{\alpha}^{k}_{2}\dots tan_{\Delta t }^{-}\dot{\alpha}^{k}_{n}]^{T}\)--Estimated vector of negative mean derivative functions by the E-TUNI at time \(t_{k}\). \(tan^{k}\dot{\alpha}^{i}=tan^{+}\dot{\alpha}^{i}=[tan_{\Delta t}^{k}\dot{\alpha} ^{i}_{1}\quad tan_{\Delta t}^{k}\dot{\alpha}^{i}_{2}\dots tan_{\Delta t}^{k} \dot{\alpha}^{i}_{n}]^{T}\)--Vector of positive instantaneous derivatives at time \(t_{k}\).