**5. Results and discussion**

The goals of this chapter are to use the ANFIS model to model, simulate and predict the behavior of *D* mesons ratios production cross-section (*D*þ*=D*0, *D*<sup>∗</sup> <sup>þ</sup>*=D*0, *D*<sup>þ</sup> *<sup>s</sup> =D*<sup>0</sup> *and D*<sup>þ</sup> *<sup>s</sup> =D*þ) and differential production cross section of prompt (*D*0, *D*þ, *D*<sup>∗</sup> <sup>þ</sup> and *D*<sup>þ</sup> *<sup>s</sup>* mesons) as a function of *PT* in pp. collisions at ( ffiffi *s* p = 5.02 and 7 TeV).

**Figure 4.** *Flowchart of ANFIS model.*

*Artificial Intelligence Approaches for Studying the* pp *Interactions at High Energy… DOI: http://dx.doi.org/10.5772/intechopen.111552*

Specific steps are taken in order to achieve the chapter's objectives. The ANFIS model is loaded with experimental data. MATLAB is utilized to process the modeling (R2017 a). Eight ANFIS networks have been generated to finish the modeling process for*D* mesons ratios production cross-section (*D*þ*=D*0, *D*<sup>∗</sup> <sup>þ</sup>*=D*0, *D*<sup>þ</sup> *<sup>s</sup> =D*<sup>0</sup> and *D*<sup>þ</sup> *<sup>s</sup> =D*þ) and differential production cross section of prompt (*D*0, *D*þ, *D*<sup>∗</sup> <sup>þ</sup> and *D*<sup>þ</sup> *<sup>s</sup>* mesons) as a function of *PT* in pp. collisions at ( ffiffi *s* p = 5.02 and 7 TeV). For each ANFIS network, the number and the type of membership functions (MF) for both inputs and outputs are determined. The number of epochs is also adjustable to obtain the lowest training error. After re-training multiple ANFIS networks with various specifications, the optimal ANFIS network with the lowest possible training error is discovered. The input patterns of the designed ANFIS hybrid have been trained to produce target patterns that modeling *D* mesons ratios (*D*þ*=D*0, *D*<sup>∗</sup> <sup>þ</sup>*=D*0, *D*<sup>þ</sup> *<sup>s</sup> =D*<sup>0</sup> and *D*<sup>þ</sup> *<sup>s</sup> =D*þ) and differential production cross section of prompt (*D*0, *D*þ, *D*<sup>∗</sup> <sup>þ</sup>, *D*<sup>þ</sup> *<sup>s</sup>* mesons) as a function of *PT* in pp. collisions at ( ffiffi *s* p = 5.02 and 7 TeV).

The best ANFIS network for this dataset was found within 50 epochs and two hidden layers with 6 and 6 neurons as specified in **Figure 5**. Membership functions (MFs) also play a fundamental role in Fuzzy Inference Systems modeling. It is worth mentioning that these system modeling are chosen carefully. The performance of ANFIS is very sensitive to the amount of MFs in the system. As a result, it is expected that complex systems with a great quantity of MFs will perform poorly because of the amount of the premise parameters that need to be estimated. MFs can have many shapes. Accordingly, the results of the experiments show that ANFIS with Generalized bell-shaped (gbellmf) has attained the best performance in all simulated and predicted experimental data (see **Figure 6**). The generalized bell-shaped MF (gbellmf) has a

**Figure 5.** *The ANFIS architecture of the best network.*

**Figure 6.** *The generalized bell-shaped membership function (gbellmf).*

bell-like symmetrical shape. According to: *<sup>f</sup>* <sup>¼</sup> ð Þ¼ *<sup>x</sup>*, *<sup>a</sup>*, *<sup>b</sup>*,*<sup>c</sup>* <sup>1</sup> <sup>1</sup><sup>þ</sup> *<sup>x</sup>*�*<sup>c</sup> <sup>a</sup>* j j<sup>2</sup>*<sup>b</sup>* . This function has three parameters: a determines the width of the bell-shaped curve, with a larger value resulting in a wider membership function, b is a positive integer that defines the shape of the curve on either side of the central plateau, and c determines the center of the curve in a universe of discourse. The function is used to generate an initial output FIS matrix from training data, with default values for membership function numbers "6 6" and types "These defaults" provide membership functions on each of the two inputs. The generated fuzzy inference system structure has 36 fuzzy rules of type "linear" by default. The following are the final parameters that best fit the data.
