## 3 Solution Methodology
### Training and Testing Database  
The generated database comprises a comprehensive set of samples to cover various scenarios encountered in the cone and disk apparatus. It includes data from experiments or simulations representing different configurations such as stationary disk and rotating cone, stationary cone and rotating disk, and combinations thereof. The database encompasses  
Figure 2: Neural network diagram for TRHNF-DC.  
a significant number of samples to ensure thorough coverage of the parameter space. Data are generated each time by varying 1 parameter and keeping other parameters fixed. Inputs to the database typically include variables such as fluid properties, geometrical dimensions of the apparatus, rotational speeds, and operating conditions. Outputs from the database consist of observed or simulated behaviors of interest, such as velocity profiles and temperature profiles. These outputs are essential for training the neural network to predict system responses accurately. By encompassing diverse configurations and parameter ranges, the database ensures that the trained neural network can effectively generalize to unseen cases and accurately predict outcomes for various operating conditions encountered in practical applications of the cone and disk apparatus. The Levenberg-Marquardt (LM) algorithm is a popular optimization method used for training neural networks, particularly because of its efficiency and speed in converging to a solution. When allocating data for training, validation, and testing in the context of the LM neural network, the rationale involves several key considerations. The data are typically split into three sets: training, validation, and testing. The training set is used to adjust the weights and biases of the network, the validation set is used to tune hyperparameters and prevent overfitting, and the testing set is used to evaluate the final model's performance. This allocation ensures that the model is both trained well and generalizable. The LM algorithm is powerful and can easily overfit the training data. By setting aside a validation set, the model's performance on unseen data can be monitored during training. If the validation error starts to increase while the training error decreases, it indicates overfitting, and training can be stopped early (early stopping). The test set, which is not used during training or validation, provides an unbiased evaluation of the final model's performance. This ensures that the model's accuracy and generalization capabilities are accurately assessed. The validation set is crucial for fine-tuning the hyperparameters (such as learning rate, network architecture, and regularization parameters). The LM algorithm benefits from this as it helps in finding the optimal balance between underfitting and overfitting. During the training process, the LM algorithm can use the validation set to monitor the network's performance at each iteration. This helps in deciding when to stop training to avoid overfitting, known as early stopping. The allocation of a validation set is particularly important for this purpose.  
The values of parameters are \(M=0,2,4\), \(\phi_{Al_{2}O_{3}}=0.02\), \(\phi_{Cu}=0.02\), \(Re_{\omega}=0.3\), \(Re_{\Omega}=0.13\), \(\eta=2\), \(n=1\), and \(Pr=6.2\) and the relevant properties of alumina, copper and water are shown in Table 1.