## 3 Results and Discussion
### ANN Model  
The experimental results obtained from our adsorption studies were used to build and train various ANN models to assess urea recovery from human urine using fecal ash. Different second-order methods were employed to train our network. The prediction accuracy and convergence time were then compared: Bayesian regularization techniques such as ANN, quasi-Newton, Levenberg-Marquardt, and conjugate gradient were considered, and the one that was the best match for our task was chosen. Out of the selected models, only two (TRAINBR and TRAINLM) exhibited considerable output performance for our study. Hence, the other models were disregarded as their performance was marginal to fit in our study. The datasets used in this study were segmented and normalized as input data for the various ANN models. This proved beneficial as none of the models in our study exhibited instances of overtraining and faced minimal computation-related problems while training the ANN models. The study results indicated that incorporating neurons in the hidden layer led to an improvement in performance and the efficiency of prediction. The actual values TN and TP and the predicted values obtained from MLR were compared to validate and assess the prediction accuracy of each model and compute the goodness of fit for each of them. The error values were calculated with absolute average deviation (AAD), adjusted-R\({}^{2}\), R-squared (R\({}^{2}\)), and root mean square error (RMSE). The residual errors obtained from the abovementioned methods were closer to zero for two models (TRAININLM and TRAINBR). This demonstrates favorable compatibility among the actual values and the models' outputs. In general, for a model to be deemed as the best model, it must exude minimal error values and highest counts in terms of correlation coefficients. Similarly, in our study, the best structure of the ANN model was decided based on the minimal error approach. Hence, BRANN was considered as a superior ANN model in this study. Cross-validation is no longer necessary with Bayesian regularized artificial neural networks (BRANNs) (Figure 2a). This aided in the validation set selection and network design optimization processes. Additionally, BRANNs may be utilized with automatic relevance determination (ARD) of the input variables, allowing the network to \"guess\" the significance of each input further. This avoids the inclusion of unnecessary or highly correlated indices in the modeling process and identifies the most critical factors for generating the activity data. Training performance using optimization is executed for 1000 epochs. Based on the mean squared error, the best training performance is achieved at 144 epochs, with an optimized value of 1.0077 for Bayesian regularization (Figure 2b). Obtaining the variations using the gradient method based on the gain leads to the error being minimized. Step by step, we can analyze the rate of change, thereby regularizing the parameters (Figure 2c). Error histograms with 20 bins were used (Figure 2d). It was seen that by using the Bayesian network, the error value could be reduced to zero. The autocorrelation of error with variations of lag leads to a minimized error with the confidence limits, as shown in Figure 2g. Responses of the outputs using Bayesian regularization are recorded with zero error, and accuracy of R = 0.9982 was recorded with a convergence time of 7 s (Figure 2e,f). The Levenberg-Marquardt algorithm, also referred to as the damped least-squares technique, was developed especially for loss functions that take the form of a sum of squared errors. It is valid even if the precise Hessian matrix is not computed--instead of that, it utilizes the gradient vector and the Jacobian matrix. A neural network is designed to analyze the performance of mean squared error using Levenberg-Marquardt (Figure 3a). For the given model, the best validation performance is obtained at epoch 1, and the value is 31.52 (Figure 3b).