**4. Discussion of Results**

The six scenarios of TD-PNFM by the variation of Prandtl fluid number, flexible number, ratio parameter, Prandtl number, Biot number and thermophoresis parameter are formulated for four different cases for both, temperature and concentration profile of three-dimensional Prandtl nanofluid flow model as elaborated in Table 1.

The Prandtl number described as the ratio of momentum to thermal diffusivity. As the Prandtl number increases, the temperature decreases. As *Pr* increases, thermal diffusivity reduces, resulting in a drop in temperature. The Biot number represents the ratio of inner to outside thermal resistance for a solid item that transfers heat to a liquid flow. This ratio indicates if the inside temperature of a body fluctuate considerably in space when it heated or cools over time due to a thermal gradient given to its sheet. A rise in the Biot number induces intense convection, resulting in an increase in thermal profile. Thermophoresis is a force caused by the difference of temperature among the cold wall and hot gas that causes particulate particles to migrate more toward the cold wall. This variable is caused by nanoparticles. Nanoparticles increased the nanoliquids thermal conductivity. The thermal conductivity of nanoliquids enhances with temperature. As a result, an increase in temperature is noticed for a better estimation of *Nt.*

The dataset for the designed TLM-BANN is computed with the help of Adam numerical method, for which, the input lies between 0 and 8 with 0.08 step size for all the four cases of six different scenarios of TLM-BANN of TD-PNFM. The solution for three-dimensional Prandtl nanofluid flow is determined by using the command 'nftool' in MATLAB. The dataset is generated for 101 points in which 81 points are for training, 10 points for testing and 10 points for validation of designed TLM-BANN.

The performance, state transition, fitness, error histogram and regression for the case I of the scenarios I, II, III, IV, V and VI are shown in Figures 4–9, respectively. The MSE based performance reflects the difference between observation and simulation; the lower the MSE value, the higher the performance. The performance is 10<sup>−</sup>10, 10−10, 10−10, 10−11, 10−<sup>10</sup> and 10−<sup>10</sup> with epochs 204, 192, 143, 20, 183 and 176, as depicted in the performance

graphs. The histogram displays the technique's dependability by displaying the difference between the anticipated and targeted results after neural network training. A histogram plot has twenty vertical bars, which are referred to as bins. The data is distributed evenly between positive and negative components is shown by the zero line, which is the red vertical line in the histogram plot. The regression plots demonstrate the output and the target relationship. Regression R measures that how accurately measured values fit a straight line or curve, and the straight line reflecting the perfect fit. If R = 1, the outputs and the targets relation is accurate. The fitness graph depicts the relation of training, testing, and validation of output and training, where overlap of all three indicating that algorithm is properly trained and provides an accurate answer. Inside the training state, there is a portion of error plot that represents the error related to output. Moreover, the data for MSE, performance, epochs, Mu and time taken is given in Table 2. The Mu and gradient values corresponding to epoch reflect whether the convergence is gradual or rapid; as the epoch advances, the values of gradient and Mu fall, indicating the convergence is quick using the Levenberg-Marquardt solver.

**Figure 4.** *Cont*.

**Figure 4.** Scenario I Case I of TD-PNFM. (**a**) MSE Results: Scenario I Case I; (**b**) Transition state: Scenario I Case I; (**c**) Fitness: Scenario I Case I; (**d**) Error Histogram: Scenario I Case I; (**e**)regression: Scenario I Case I.

**Figure 5.** *Cont*.

**Figure 5.** Scenario II Case I of TD-PNFM. (**a**) MSE Results: Scenario II Case I; (**b**) Transition state: Scenario II Case I; (**c**) Fitness: Scenario II Case I; (**d**) Error Histogram: Scenario II Case I; (**e**)regression: Scenario II Case I.

**Figure 6.** *Cont*.

**Figure 7.** *Cont*.

**Figure 7.** Scenario IV Case I of TD-PNFM. (**a**) MSE Results: Scenario IV Case I; (**b**) Transition state: Scenario IV Case I; (**c**) Fitness: Scenario IV Case I; (**d**) Error Histogram: Scenario IV Case I; (**e**)regression: Scenario IV Case I.

**Figure 8.** *Cont*.

**Figure 8.** Scenario V Case I of TD-PNFM. (**a**) MSE Results: Scenario V Case I; (**b**) Transition state: Scenario V Case I; (**c**) Fitness: Scenario V Case I; (**d**) Error Histogram: Scenario V Case I; (**e**)regression: Scenario V Case I.

**Figure 9.** *Cont*.

**Figure 9.** Scenario VI Case I of TD-PNFM. (**a**) MSE Results: Scenario VI Case I; (**b**) Transition state: Scenario VI Case I; (**c**) Fitness: Scenario VI Case I; (**d**) Error Histogram: Scenario VI Case I; (**e**) regression: Scenario VI Case I.



The solution for three-dimension Prandtl nanofluid model for the temperature profile is depicted in Figure 10a,c,e,g,i,k, whereas the AE analysis plots for the temperature profile is given in Figure 10b,d,f,h,j,l. Similarly the solution plots for the concentration profile is shown in Figure 11a,c,e,g. The AE analysis plots are given in Figure 11b,d,f,h. The solution plots for the velocity distribution are shown in Figure 12a,b. The outcomes for the skin friction and Nusselt number are depicted in Figures 13a–c and 14a, respectively.

**Figure 10.** Assessment of TLM-BANN for *θ* with reference dataset of TD-PNFM (**a**) Variation of *β*<sup>1</sup> for *θ*; (**b**) AE analysis in variation of *β*<sup>1</sup> for *θ*; (**c**) Variation of *β*<sup>2</sup> for *θ*; (**d**) AE analysis in variation of *β*<sup>2</sup> for *θ*; (**e**) Variation of *α* for *θ*; (**f**) AE analysis in variation of *α* for *θ*; (**g**) Variation of *γ* for *θ*; (**h**) AE analysis in variation of *γ* for *θ*; (**i**) Variation of Pr for *θ*; (**j**) AE analysis in variation of Pr for *θ*; (**k**) Variation of *Nt* for *θ* (**l**) AE analysis in variation of *Nt* for *θ*.

**Figure 11.** *Cont*.

**Figure 11.** Assessment of TLM-BANN for *ϕ* with reference dataset of TD-PNFM. (**a**) Variation of *β*<sup>1</sup> for *ϕ*; (**b**) AE analysis in variation of *β*<sup>1</sup> for *ϕ*; (**c**) Variation of *β*<sup>2</sup> for *ϕ*; (**d**) AE analysis in variation of *β*<sup>2</sup> for *ϕ*; (**e**) Variation of *α* for *ϕ*; (**f**) AE analysis in variation of *α* for *ϕ*; (**g**) Variation of *Nt* for *ϕ*; (**h**) AE analysis in variation of *Nt* for *ϕ*.

**Figure 12.** Assessment of TLM-BANN for *f* with reference dataset of TD-PNFM. (**a**) Variation of *β*<sup>1</sup> for *f*; (**b**) Variation of *β*<sup>2</sup> for *f*.

**Figure 13.** Assessment of TLM-BANN for *f* with reference dataset of TD-PNFM. (**a**) Variation of *Ha* for *Cf* Re1/2 *<sup>x</sup>* ; (**b**) Variation of *α* for *Cf* Re1/2 *<sup>x</sup>* ; (**c**) Variation of *α* for *Cf* Re1/2 *<sup>y</sup>* .

**Figure 14.** Assessment of TLM-BANN for *f* with reference dataset of TD-PNFM. Variation of *Nt* for *Nux*Re−1/2 *<sup>x</sup>* .
