# Mathematical Modeling to Estimate Photosynthesis: A State of the Art
### Mathematical Modeling  
Any branch of science, as it progresses from qualitative to quantitative, is likely to reach the point where the use of mathematics to connect experiment and theory is essential. Mathematical modeling consists of the following steps [44]:  
1. Definitions;
2. Systems analysis;
3. Modeling;
4. Simulation;
5. Validation. Mathematical models can be classified into mechanistic (white box), empirical (black box), and hybrid (gray box). These, in turn, have sub-classifications, as shown in Figure 1[36]. \begin{table}
\begin{tabular}{c p{142.3 pt}} \hline \multicolumn{2}{c}{**Methods Used for**} & \multicolumn{1}{c}{**Description**} \\ \multicolumn{2}{c}{**Photosynthesis Estimation**} & \\ \hline Gas exchange & \begin{tabular}{c} Consists of isolating the sample for analysis in a closed chamber to quantify the CO\({}_{2}\) concentration [29; 31]. Concentrated CO\({}_{2}\) gas is detected by an infrared gas sensor (called IRGA for Infra-Red Gas Analysis sensors) [32] \\ \end{tabular} \\ \hline Carbon isotopes & \begin{tabular}{c} Uses carbon isotopes such as \({}^{11}\)C, \({}^{12}\)C, and \({}^{14}\)C to produce incorporated CO\({}_{2}\) with radioactivity. This methodology is applied to analyze samples in isolated and illuminated chambers to produce a maximum fixation of radioactive CO\({}_{2}\) during photosynthesis [33; 34]. The main disadvantage is that it is destructive as it fixes a radioactive compound onto the sample; and its precision depends on lighting conditions. \\ \end{tabular} \\ \hline Acoustic waves & \
\begin{tabular}{c} Based on the principle of sound wave distortion in the medium in which waves propagate. The technique involves placing an acoustic transmitter on the seabed area where you want to monitor photosynthetic activity. The disadvantage is that it dependent on water conditions and sensitive to environmental disturbances [35]. \\ \end{tabular} \\ \hline Fluorescence &
\begin{tabular}{c} Way in which a certain amount of light energy absorbed by chlorophylls is dissipated. The fluorescence emission can be analyzed and quantified, which provides information on the electron transport rate, the quantum yield and the existence of photoinhibition of photosynthesis. Indeed, fluorescence is used in various ways, and it has different applications. The interested reader is referred to reference [5; 8; 24]. \\ \end{tabular}
\end{table} 
Table 1: _Cont._Empirical models, also called black box models, mainly described a system's responses by using mathematical or statistical equations without any scientific content, restrictions, or scientific principle. Depending on particular goals, this may be the best type of model to build [44]. Its construction is based only on experimental data and does not explain dynamic mechanisms; this refers to the fact that the system's process is unknown [40]. Estimating an unknown function from the observations of its values is a problem. The basic advice in this aspect is to estimate models of different complexity and evaluate them using validation data. A good way to restrict certain classes of models' flexibility is to use a regularized fit criterion. A key issue is finding a sufficiently flexible parameterization model. Another key is to find a suitable \"close approach \"to the model structure [45]. Researchers usually employ methods for predicting physiological parameters by using intelligent algorithms, such as Support Vector Machines (SVM), Back-Propagation Neural Network (BPNN), Artificial Neural Network (ANN), Deep Neural Network (DNN), and the combination of Wide and Deep Neural Network (WDNN) [46]. Mechanistic models, also called white box models, provide a degree of understanding or explanation of the modeled phenomena. The term \"understanding\" implies a causal relationship between quantities and mechanisms (processes). A well-built mechanistic model is transparent and open to modifications and extensions, more or less without limits.