# The Euler-Type Universal Numerical Integrator (E-TUNI) with Backward Integration
## 8 Conclusions  
As far as we know, all previous work on E-TUNI was developed using only forward integration. Therefore, we present here, for the first time, a work that uses E-TUNI in a backward integration process. Three case studies were carried out. Non-linear and autonomous ordinary differential equations govern all three dynamical systems considered in this study.  
The Euler-type Integrator designed with instantaneous derivatives is not very accurate. However, the E-TUNI has numerical precision equivalent to that of the highest-order integrators. It is interesting to note that the elegance of the theory of ordinary differential equations can be improved when combined with the theory of universal approximator of functions. This propriety is possible because artificial neural networks allow working with real-world input/output training patterns. This fact enables refining the final solution obtained by the first-order Euler integrator and estimator. Because of this, the E-TUNI can also be used to predict real-world dynamics (e.g., river flow and stock market forecasts, among others).  
Additionally, we can call the forward mean derivatives functions positive mean derivatives functions. Equivalently, we can call the backward mean derivatives functions negative mean derivatives functions. Thus, we also conclude that the mean derivative functions depend on the integration step and are asymmetric. This propriety means training two distinct neural networks to simultaneously perform forward and backward integration. Unfortunately, the positive and negative instantaneous derivative functions are symmetric to each other.  
Additionally, it would be interesting to carry out a future study of E-TUNI using deep neural networks instead of shallow neural networks, aiming to reach two essential points: (i) to be able to increase the number of training patterns (and, consequently, the domain of state variables) and (ii) try to reduce the training mean squared error, as this quantity is critical, in the case of modeling non-linear dynamic systems designed through artificial neural networks. Interestingly, E-TUNI is an excellent option to replace the NARMAX model and even the Runge-Kutta Neural Network (RKNN).  
Finally, as another future contribution, one could consider changing the notation used in this article to something more straightforward, as explained in [34; 35]. It is essential to notice that the same notation used in articles [16; 26; 27; 28; 31] was used here. In this sense, the most significant criticism would be not using the over-index \(i\) in the algebraic development of E-TUNI. However, in [31], the over-index \(i\) was used to help demonstrate the general expression of E-TUNI. Furthermore, E-TUNI could also be tested for dynamic systems using control variables.  
**Author Contributions:** [PERSON] designed the proposed model and wrote the main part of the mathematical model. [PERSON], [PERSON] and [PERSON] supervised the writing of the paper and its technical and grammatical review. [PERSON] and [PERSON]. (supervised by [PERSON], [PERSON] and [PERSON]) developed the software and performed several computer simulations to validate the proposed model. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
**Acknowledgments:** I would like to thank my great friend [PERSON] for his valuable tips for improving this article. Finally, I would also like to thank the valuable improvement tips given by thegood reviewers of this journal. The authors of this article would also like to thank [PERSON] for making all of this possible.  
The authors declare no conflicts of interest.