# Fault Diagnosis via Neural Ordinary Differential Equations
## 7 Conclusions  
Over the past few years, the development of new techniques for deeper networks such as ResNet has inspired the training of neural networks using ODE solvers as a manner to generate a continuous layer representation and even further allowed to improve the representation of systems dynamics by neural ODE networks. Therefore, a hybrid approach was presented in this article for fault diagnosis of sensors, actuators, and process faults. Results of simulations show that high performances of fault diagnosis can be obtained using neural ODE networks not requiring extensive data-processing and advanced modeling. In a nonlinear benchmark system, the training efficiency was demonstrated for each fault type, and shows an improvement when compared with classical Deep Learning models. Here, the accuracy of the network is demonstrated to be higher than 90%. In addition, a visual representation is proposed by using phase diagram representations. Likewise, the data needed to train effectively the network is reduced which permits to avoid elaborated procedures such as feature extraction, pre-processing and data augmentation.  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Model** & **Accuracy** & **Weighted Accuracy** \\ \hline Support Vector Machine & & \\ (Polynomial Kernel) & 93.3\% & 62.657\% \\ \hline Artificial Neural Network & & \\ (4 Fully connected Layers) & 80.9\% & 74.8\% \\ \hline Convolutional Neural Network & & \\ (2 Convolutional Layers + 1 Fully connected Layer) & 77.5\% & 55.8\% \\ \hline Recurrent Neural Network & & \\ (4 LSTM Layers + 1 Fully connected Layer) & 80.53\% & 73.8\% \\ \hline Neural ODE \& Dense Network & \\ (Neural ODE + 2 Fully connected Layers) & 98.5\% & 90\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Accuracy for different Machine learning Models.  
**Author Contributions:** All the authors contributed to the design of the model, the result analysis, and the writing and review of the paper. Specifically, [PERSON] was in charge of doing the simulation and training of the network, and also of the evaluation of the FDI system in the benchmark system, [PERSON] of the introduction and state of the art and [PERSON] and [PERSON] of the overall ideas of the exposed research and the general conception of the paper. 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 data presented in this study is available on request from the corresponding author.  
**Acknowledgments:** The authors are grateful for the finantial support from Proyecto Concytec--Banco Mundial \"Mejoramiento y Ampliacion de los Servicios del Sistema Nacional de Ciencia Tecnologia e Innovacion Tecnologica\" 8682-PE, a traves de su unidad ejecutora Fondecyt, and from Contrato de Adjudicacion de fondos N\({}^{\circ}\) 10-2018-FONDECYT/BM-Programas de Doctorados en Areas Estrategicas y Generales.  
**Conflicts of Interest:** The authors declare no conflict of interest.