# Predicting the Compressive Strength of Concrete Using an RBF-ANN Model
## 5 The Results of Training and Testing  
The computer program is coded with Fortran 90 and compiled with the X64 compiler of Intel(r)-Fortran 2016. Each run uses only one processor of the computer of Intel(r) Core(tm) i7-8700 CPU @3.20 GHz 3.19 GHz. Artificial neural networks with 2-5 RBFs are trained and tested. Each RBF-ANN is trained with three runs in which \(2\times 10^{8}\) iterations are processed. The computational time does not increase linearly with the number of RBFs. For the ANNs with 2 RBFs, the computational time is 2.98 h. As per those with 3, 4, and 5 RBFs, the computational times are 4.25, 5.56, and 6.81 h respectively. Though the centers of RBFs and their shape parameters are all set up with random numbers, after \(2\times 10^{8}\) iterations of training, the ANNs with the same number of RBFs converge to results that are very close to each other. That means the convergence is valid for sure. It also confirms the presumption that using the stochastic gradient approach is enough. For the ANNs with just 2 or 3 RBFs, converging to the final result just needs several million times of iteration, which means the convergence is achieved within 1 hour.  
The root-mean-square error is employed to examine the performance of the trained ANNs.  
\[E_{r.m.s.}=\sqrt{\left(\eta-\hat{\eta}\right)^{2}} \tag{20}\]  
where the overbar represents the average in the data set. Note that the numbers of data in the training set and testing set are 499 and 72, respectively. The results of training and testing are plotted in Figure 1.  
It is found that setting more RBFs in the ANN results in smaller \(E_{r.m.s.}\) in the training set. However, it does not go the same way in the testing set. This implies too many RBFs in the ANN have a risk of over-fitting. Though the ANN with 4 RBFs fits testing data the best, using just 3 RBFs is supposed to be safer from over-fitting. The comparisons of exact values and the predicted values by the ANN with 3 RBFs are shown in Figure 2. Note that the plotted values in this figure are the normalized values. The root-mean-square errors of the training set and the testing set are 0.0822 and 0.0895, respectively. It is worthy to note that the testing set is the same as in [27] while the RBF-ANN in this study fits the testing set better. The root-mean-square error of the testing set in [27] was 0.0914.  
Figure 1: The performance of the trained radial basis functions-artificial neural networks (RBF-ANNs).  
Figure 2: The comparison of the outputs of ANN with their target values in which the ANN only has 3 RBFs.  
Finally, the centers of the 3 RBFs are listed here.  
\[\zeta_{1}=\left[\begin{array}{cccccc}-1.00000&-1.00000&0.44034&0.60326&-0.00486&0. 52630&1.24615\end{array}\right]^{T} \tag{21}\]  
\[\zeta_{2}=\left[\begin{array}{cccccc}-0.84767&0.61160&0.46294&0.56556&0.67973&0. 61628&0.65405\end{array}\right]^{T} \tag{22}\]  
\[\zeta_{3}=\left[\begin{array}{cccccc}-0.55917&1.05027&1.08523&0.79403&0.86121& 0.77691&0.46370\end{array}\right]^{T} \tag{23}\]  
The shape parameters of these 3 RBFs are 2.06819, 0.90677, and 0.45945, respectively. Their weights are \(-4.17952\), \(11.2345\), and \(-5275.82\) while the threshold is 1.05900. With these numbers, one can use Equations (7) and (13) to calculate the corresponding output of the ANN by any input. The 28-day compressive strength of the concrete can be retrieved by using the backward transform of Equation (4).