# Fault Diagnosis via Neural Ordinary Differential Equations
## 1 Introduction  
Fault diagnosis systems are of great importance in industrial processes, especially in those with high stakes systems, where it is important to perform continuous diagnostics and monitoring of the main variables. Furthermore, it could also be used to perform fault prediction for improving the maintenance plan and to avoid undesired malfunctions. In the fault detection literature, a great number of methods have been dedicated to model-based techniques [1, 2] with great success in many cases, although a known drawback of such methods is their limitation to processes with relatively simple dynamics and sensiblity to model approximations. Examples of model-based methods are: the structural approach, which searches for irreducible relations that attains sensibility to the selected faults [3, 4], and eigenstructure-based fault diagnosis, which deals with a minimization problem for the sensibility of perturbations against faults [5, 6]. The same minimization problem is approached using Linear Matrix Inequalities (LMI) [7], and another common approach is to use observers, such as sliding mode observers [8], Kalman-based fault detection [9], and also residuals generation using parity space relations [10, 11]. Likewise, for complex processes with functional constraints, methods have been developed that consider decentralized and distributed diagnostic architectures [12, 13]. Another important group of fault detection techniques, regarded as data-based approaches, are used when the model of the plant under evaluation is not available. They are dedicated to the analysis of collected data to detect unusual trends in it, and thus identify faults in the system. Principal Component Analysis (PCA) performs a dimensionality reduction of the data map for finding faults in the system, although it requires high amounts of data [14, 15]. Support Vector Machines (SVM) is a technique for classification that performs high generalization even with small data-sets but can suffer from data imbalance, although learning ability can be added such as in the multikernel SVM [16]. Artificialneural networks (ANN) are employed using supervised training and can detect faults after learning hidden relations in the data [17]; however, they lack visual representation and can suffer from overfitting. Use of Deep Learning (DL) techniques have become increasingly popular in the latter years due to an increase in power computation, higher availability of information, and high accuracy, many DL models have been tested such as Dense Neural Networks [18], Deep Belief Networks [19; 20; 21], Convolutional Neural Networks [22; 23], Recurrent Neural Networks [24; 25], and Generative Neural Networks [26], however, in most of the examples the DL model is adapted only to the plant in the study, therefore a change of plant will require a change of the methodology. Some works have also been devoted to the elaboration of hybrid techniques, model-based plus data-based systems for fault diagnosis. These techniques aim to have combined advantages from both previously mentioned approaches, they use mainly machine learning techniques for fault identification coupled with model-based fault detectors. In [27], a hybrid model-based and data-driven system is elaborated for fault detection and diagnosis, although both methods are used independently. In [28], [PERSON] proposed the use of SVM to improve fault identification, and in [29], the merging mechanism between residuals and SVM is improved for fault location. In [30], the hybrid mechanism is used for the selection of residual generators. Meanwhile, in [31], a Modified PCA was combined with filters to detect faults. On the other hand, the use of DL techniques in hybrid fault diagnosis systems has not been extensively developed, although the use of DL is recognized for improving accuracy in many other recognition tasks [32].