II. Neuronal Structure

B. Data Scale

We, as authors, chose to perform the neural model with the actual data without scaling. Table 3 establishes that the statistical criteria are more favorable for the unscaled data than the scaled data from the logarithm in base 10.

C. Artificial Neural Network (ANN) Architecture

Due to the nonlinearity of the functions that estimate the diameter (12) and the friction coefficient (14), it is feasible to estimate the output parameter through optimization algorithms. Artificial neural networks can approximate any continuous nonlinear function independent of the function degree [⁴]. The implementation of artificial intelligence techniques contributes to finding the minimum error surface generated by the cost function. A neural structure created in Matlab (2021a) is proposed (Figure 3) corresponding to 6 input signals (Q, H, L, ε, 𝜗, Ʃk), a hidden layer with 25 neurons, and an output signal for diameter estimation. The logsig transfer function showed optimal results in terms of MSE and computational time required to reach convergence. The Levenberg-Marquardt (trainlm) method is suited to both the training data and the test data sets. The network weights are iteratively adjusted from the error estimate [⁵] [⁶] describes the application of the Levenberg-Marquardt in neural network systems for training. This algorithm has demonstrated a higher training speed for the neural network [⁷]. Similarly, [⁸] proposes a neural architecture of 5 input variables, a hidden layer with 36 neurons, and 10 output parameters to classify the optimal commercial diameter for the hydraulic system.

Fig. 3 Schematic diagram of neuronal architecture (6-25-1). 

Table 4 indicates different neural structures tested for this study to obtain the lowest value for the MSE; this value was achieved for the architecture (6-25-1) with an MSE equal to 5.41E-6. It was found that increasing the number of hidden layers does not guarantee a decrease in the MSE, and as a consequence, it does increase the computational cost of the iterative process. The computational time for the (6-25-25-25-1) scheme was 3 hours and 25 minutes. In contrast, the time required for the (6-25-25-1) scheme was 15 minutes. Thus, neural models with several hidden layers tend to overfit so that the model can predict the training data. However, for the prediction of independent data, the overfitted neural model has shortcomings. The most important property of a neural network is its ability to generalize and model new input data [⁹].

D. Neural Network Training

The Levenberg-Marquardt training function (trainlm) uses the second derivatives of the cost function upgrading the convergence times [¹⁰]. The implementation of this algorithm is feasible as long as the second derivative of the neural network weights exists. Thus, the input signals are affected by the random weights and biases values (16). Once this value is obtained, the log-sigmoid activation function (logsig) (15) is implemented. This function implements the Jacobian matrix for the calculations. This matrix is formed by the first-order partial derivatives of the function. The performance for this function is measured through the MSE. The algorithm presents good performance for escaping moderate local minima and oscillation problems. The interval [0,1] determines the range of the corresponding function.

Fig. 4 Log sigmoid transfer function. 

The following expression defines the log-sigmoid activation function (logsig):

Six neural architectures are proposed for the diameter in pressure piping systems calculation. The lowest Pearson Correlation Coefficient was obtained for the scheme (6-2-1) with an R equal to 0.99282; the computational time required for this scheme was 4 minutes. Figure 5a presents the dispersion of the outputs for ANN (6-2-1). However, for the arrangement (6-25-1), represented in Figure 5d, an R equal to 0.99939 indicates the best training and output signals fit; the computational time required was approximately 10 minutes. The scheme (6-30-30-1) presented a similar performance to the system (6-25-1). Nevertheless, this scheme composed of two hidden layers considerably increased the computational time required to reach convergence, approximately 3 hours and 25 minutes.

Fig. 5 ANN architectures. 

Figure 6 shows the 3D irregular surface structured from the weights, the bias parameter, and the sum of squared errors. The objective of the cost function is to establish the minimum for this surface.

Fig. 6 Error surface for ANN (6-25-1). 

Figure 7a indicates the performance of the validation and training curve; the error decreases as the number of epochs of the iterative process increases. If the MSE of the validation curve starts to increase and distance itself from the training curve, the model will overfit, which directly affects the generalization capability of the network. Figure 7b presents the adaptation parameter mu used in the Levenberg-Marquardt optimization process. For this study, the mu parameter corresponds to 1E-08 for season 68.

Fig. 7 Training status (6-25-1). 

The data were divided into three sets using random indexes, training, validation, and test, as represented in Figure 8. The straight line at 45° represents a perfect fit, i.e., the values of the estimated outputs are equal to the target values. Figure 8c indicates the relationship between outcomes and targets for the test set. This study found a linear relationship between the output signals and the objectives with a Pearson Correlation Coefficient equal to 0.99946 for a neural scheme (6-25-1).

Fig. 8 Training regression (6-25-1). 

E. Validation

For the analysis of the network output signal and to validate the independent data set, six statistics were considered: R (Pearson correlation coefficient), MAE (Mean Absolute Error), MSE (Mean Squared Errors), SSE (Sum of Squared Errors), SAE (Sum of Absolute Errors), BCE (Binary Cross Entropy). In order to establish a representative spectrum associated with the independent data set, a set of 1,000 random data was structured for the test signals. Cross-validation allows comparing the estimated signals with the target values from the independent data of the training set. The error histogram presented in Figure 7c indicates a minimum standard deviation with a mean tending to zero for the individual data; the histogram tends to be symmetric around the average. The percentage difference of the MSE between the training data and the independent data corresponds to 0.00015%, indicating the generalization capability of the neural model.

The U.S. Environmental Protection Agency developed ®Epanet for the calculation of pressurized piping systems. This system works with hydraulic simulation periods from the drinking water distribution systems. It also models water quality within a pressurized network and can be used for any non-compressible fluid flowing under pressure analysis. ®Epanet determines the flow rates through the pipes and the values of the pressures at the nodes based on the principle of conservation of mass and energy by implementing the gradient method proposed by Todini and Pilati (1987). For the calculation of friction losses, three models are presented: Darcy-Weisbach (D-W), Hazen-Williams (H-W), Chézy-Manning (C-M).

Similarly, the neural network was evaluated from the data shown in Table 6. The results obtained were compared with the values calculated in ®Epanet, ®Excel, and the application for the calculation of pressure pipes of the hydraulics online website, accepting the Darcy Weisbach equation (2). For the first data in Table 6, the modeling was performed in ®MatLab software. Two tanks were established (Figure 1), the first tank with a height above the water surface equal to 36.712 m, a fitting at the inlet with a coefficient of 1, the main pipe with a diameter of 0.2428 m, a length equal to 104.31 m, a loss coefficient per fitting equal to 1 at the outlet, a tank at the end with zero height, the kinematic viscosity corresponds to 0.000001404 m2/s, and the roughness of the pipe equals to 0.0002574 m. The model in ®Epanet obtained a flow rate equal to 0.38109 m3/s, with a velocity of 8.23 m/s. The results obtained in ®Epanet, ®Excel, and the website validate the values calculated by ANN (6-25-1). According to the hydraulic calculation performed in ®Epanet, the velocity remains constant because the system does not present a variation of the pipe cross-section. This performance causes the Reynolds number and the friction coefficient to remain constant along the length of the pipe.

In addition, the generalization capability of the neural model is evidenced by the estimation of 50 independent data. These data show no dependence on the training, test, and validation data set. Figure 10 shows the actual data (circles) obtained from equations (12), (14), and the ANN estimated data (crosses). The Pearson Correlation Coefficient obtained was equal to 9.9742E-1 showing a linear relationship between the input signals and the estimated signals. The root average squared error for the 50 data corresponded to 3.65E-05. The cross-entropy obtained was equal to 5.504E-1, which establishes a lower uncertainty for the probability distribution.

Fig. 9 Discharge - speed. 

Finally, Figure 10 shows the generalization capability of the neural network (6-25-1) to estimate the theoretical diameter in pressurized pipes. This study demonstrated the potential of artificial neural networks to solve nonlinear systems.

Fig. 10 Diameter - Head Loss. Patterns - Estimated ANNs.