## 4 Conceptual Framework for Augmented Reality Application
### Modeling
#### 4.3.2 Mathematical Models  
Mathematical models, in turn, are essential tools in industrial plants' design and operation stages. These models can be divided into data-driven models, deterministic models based on fundamental laws of chemistry and physics, and stochastic models based on probability laws. This classification is not unique and will produce different combinations depending on the objective of the categorization.  
Data-driven models have gained prominence in recent years due to the versatility of training, application, and use [49; 54; 58; 59]. Data-driven models can be divided between machine learning and artificial intelligence. In both cases, the applications are varied. On the one hand, artificial intelligence seeks to build models and algorithms that mimic human behavior in decision making. On the other hand, machine learning is a branch in which the objective is to make a machine learn to reproduce a behavior observed in the data or encouraged by the reward functions.  
In turn, phenomenological mathematical models are constructed based on fundamental laws [60; 61; 62]. The main characteristic of these models is that they have a theoretical basis and an excellent capacity for extrapolation. These models can be built based on differential or algebraic equations in the different possible applications. In the first case, the models obtained may be dynamic or not and dependent on spatial or temporal coordinates. In the second case, the models will not depend on differential terms of time or space but may explicitly depend on time or exogenous inputs. In both cases, the models can be used for applications that have or do not have an explicit dependence on time. Data-oriented and phenomenological models can be used to operate, control, and monitor production in industrial systems. However, some differences make the use distinct. On the one hand, data-driven models have a higher computational construction cost than models based on fundamental laws. On the other hand, phenomenological models are more challenging to build for complex systems and, in some specific cases, have a computational cost considerably higher than a data-oriented model.  
Data-driven models are of great importance in building digital twins for process systems. That importance derives precisely from the computational cost involved in solving phenomenological models. In some applications, the numerical solution of a complex phenomenological model may have a running time close to the actual operating time of the system. This long processing time makes implementing a phenomenological digital twin difficult. On the other hand, using a data-oriented surrogate model allows several scenarios to be tested with less computational time. Additionally, when a digital twin is used for production planning and optimization, the optimization algorithms perform numerous objective function queries and solves each model during the problem solving. If the model used is based on fundamental laws, the computational cost of these applications can become prohibitive. On the other hand, using data-oriented models makes solving these problems feasible in parallel with the real process [63].  
Another set of possible models is based on probability analysis. These models are known as stochastic. The main feature of these models is that they do not provide the same answer for the same input set, so the model prediction is obtained based on the probability of occurrence of concurrent events. Commonly, these models are used more frequently to represent financial or human systems. However, in industry, these models provide essential information in dealing with the reliability and probability of failure of instruments, equipment, and human beings in a factory site and the design and representation of queue management systems.