**ANFIS info.**

Number of nodes: 101

Number of linear parameters: 108

Number of nonlinear parameters: 36

Total number of parameters: 144.

Number of fuzzy rules: 36

Simulation and prediction results based on ANFIS used to model the*D*—meson 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>, *D*<sup>þ</sup> *<sup>s</sup>* mesons) for pp. collision at different *PT* and ffiffi *s* p (5.02 and 7TeV) are given in **Figures 7**–**10(a**–**d)**. The experimental data is represented by solid circle symbols, while the simulated ANFIS results are represented by solid line curves and the predicted ANFIS results are represented by solid square symbols. After training, the ANFIS models have been used to predict *D* meson ratios at different values of *PT* and ffiffi *s* p (5.02 and 7 TeV) that are not used in training session as shown in **Figures 7**–**10(a**–**d)**. **Figures 7** and **8(a**–**d)**, they introduce the ANFIS trained results for Transverse momentum distribution measured at ffiffi *s* p (5.02 and 7TeV) respectively for particles ratios and prediction for experimentally unmeasured values. Studying **Figures 7** and **8**, it was discovered that the simulation ANFIS curves and the experimental data symbols nearly matched, indicating excellent simulation results and the prediction data symbols are clear and produce acceptable

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

**Figure 7.** *Transverse momentum distribution measured at* ffiffiffi *S* <sup>p</sup> *=5.02 TeV for particles ratios (D*þ*=D***<sup>0</sup>**ð Þ<sup>a</sup> , *<sup>D</sup>*<sup>∗</sup> <sup>þ</sup>*=D***<sup>0</sup>**ð Þ **<sup>b</sup>** , *<sup>D</sup>*<sup>þ</sup> *<sup>s</sup> <sup>=</sup>D***<sup>0</sup>**ð Þ**<sup>c</sup>** and *<sup>D</sup>*<sup>þ</sup> *<sup>s</sup> =D*þð Þ **d** *are compared with ANFIS simulation and prediction.*

**Figure 8.** *Same as in Figure 7 but at* ffiffiffi *S* <sup>p</sup> *= 7 TeV.*

results. The ratios of the *PT*-differential cross sections of prompt mesons in pp. collisions at ffiffi *<sup>s</sup>* <sup>p</sup> = 5.02 TeV, the transverse momentum interval 0 <sup>&</sup>lt;*PT* <sup>&</sup>lt;30 GeV/c are reported in **Figure 7(a**, **b**, **d)**, and 0<*PT* < 20 in **Figure 7(c)** but in **Figure 8** at ffiffi *s* p = 7 TeV, the transverse momentum interval 1< *PT* <20 GeV/c are reported in **Figure 8(a**, **b)**, and 3< *PT* <12 GeV/c in **Figure 8(c**, **d)**. In **Figure 9**, the production cross sections were

#### **Figure 9.**

*PTdifferential production cross section of prompt <sup>D</sup>***<sup>0</sup>**ð Þ**<sup>a</sup>** *, <sup>D</sup>*þð Þ **<sup>b</sup>** *, <sup>D</sup>*<sup>þ</sup> *<sup>s</sup>* ð Þ**<sup>c</sup>** *, <sup>D</sup>*<sup>∗</sup> þð Þ **<sup>d</sup>** *mesons in* pp *collisions at* ffiffi *s* p *= 5.02 TeV using ANFIS.*

**Figure 10.** *Same as in Figure 9 but at* ffiffiffi *S* <sup>p</sup> *= 7 TeV.*

measured at ffiffi *<sup>s</sup>* <sup>p</sup> = 5.02 TeV in the transverse momentum interval 0<*PT* <sup>&</sup>lt;30 GeV/c for *D***<sup>0</sup>**, 1<*PT* <30 GeV/c for *D*<sup>þ</sup> and *D*<sup>∗</sup> <sup>þ</sup>, and in 2<*PT* < 24 GeV/c for *D*<sup>þ</sup> *<sup>s</sup>* mesons and were measured at ffiffi *<sup>s</sup>* <sup>p</sup> = 7 TeV, in the transverse momentum interval 0 <sup>&</sup>lt;*PT* <sup>&</sup>lt;<sup>30</sup> GeV/c for *D***<sup>0</sup>**, 1<*PT* <24 GeV/c for *D*<sup>þ</sup> and *D*<sup>∗</sup> <sup>þ</sup>, and in 2 <*PT* <12 GeV/c for *D*<sup>þ</sup> *s* mesons as shown in **Figure 10**. **Figures 9** and **10(a**–**d)**, they introduce the ANFIS trained results for *PT* differential production cross section of prompt mesons in pp. collisions at ffiffi *s* p (5.02 and 7TeV) respectively. The ANFIS simulation results show a *Artificial Intelligence Approaches for Studying the* pp *Interactions at High Energy… DOI: http://dx.doi.org/10.5772/intechopen.111552*


**Table 2**

*RMSE and R<sup>2</sup> for D- meson 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*) in* pp *collision.*


**Table 3**

*RMSE and R<sup>2</sup> for differential production cross section of prompt D*<sup>0</sup>*, D*þ*, D*<sup>∗</sup> <sup>þ</sup>*, D*<sup>þ</sup> *<sup>s</sup> mesons in* pp *collision.*

high level of agreement with the experimental data. We notice that the curves obtained by ANFIS hybrid model show the best fitting to the experimental data for simulation and prediction. This is agreed with Khajeh [28]; and Tortum, Yayla [29]; and G. M. Behery [30]. The Minimum root mean squared error (0.151, 0.012, <sup>2</sup>*:*<sup>07</sup> � <sup>10</sup>�7, 4*:*<sup>95</sup> � <sup>10</sup>�7, 1.0819, 0.2626, 0.1426 and 1*:*<sup>86</sup> � <sup>10</sup>�<sup>5</sup> ) and higher coefficient of correlation (0.9327, 0. 9795, 1, 1, 0.998, 0.999, 1 and 1) for ANFIS (1–8) respectively were obtained as shown in **Tables 2** and **3** respectively.
