# Prediction of Critical Currents for a Diluted Square Lattice Using Artificial Neural Networks
## 4 Simulation Results  
Our experimental setup generated a data set comprising several values of current and voltage for a wide range of magnetic field and temperature readings, a few entries of which are given in Table 1. Our methodology used 70 percent of these values as training data, while the rest was used to serve the validation and prediction purposes. We made use of MATLAB's toolbox called Neural Network to perform these simulations. As mentioned already, we implemented three architectures; each was trained using six algorithms, where each algorithm was run ten times with a different number of neurons in the hidden layers. This generated a total of 180 different models (6 algorithms \(\times\) 10 configurations \(\times\) 3 architectures). Table 2 summarizes the six algorithms and ten sets of number of neurons for each iteration of the algorithms, leading to sixty entries in total. Note that the entry No. of neurons \([x\;y]\) indicates \(x\) neurons in the first hidden layer, and \(y\) in the second. Each one of the 180 models was trained five times, and the best results in terms of minimum MSE, epochs, and training time were saved. The obtained results showed MSE in the range of \(6.5\times 10^{-6}\) to \(2.7\times 10^{-8}\). For the purpose of cross-validation, results from all of the models were compared with the actual data generated by the PPMS. Figures 4 and 5 show the actual and ANN-predicted IV curves for the diluted square array of antidots. The measurements--specifically for testing--were taken with the following parameters: temperature = 8.5 K, magnetic field = 300 Oe, and temperature = 8.4 K, magnetic field = 100 Oe, respectively, for the two figures, while current was varied from 0 to 8 mA in each case. Note that these values were deliberately not included in the training process of the ANN--they have been explicitly used for validation. Figure 6 shows MSE in IV curves predicted by 180 models. Note that the horizontal axis corresponds to the sixty entries of Table 2. It may be observed in the figure that the feedforward network with [12; 10] neurons in the hidden layers for \(BR\) as the training algorithm achieves the lowest MSE (i.e., \(4.55\times 10^{-8}\)). However there are other results that have MSE in the same range, mostly trained with \(BR\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Model** & **No. of** & **Train.** & **Model** & **No. of** & **Train.** & **Model** & **No. of** & **Train.** \\
**No.** & **Neurons** & **Algo.** & **No.** & **Neurons** & **Algo.** & **No.** & **Neurons** & **Algo.** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 2: Sixty different ANN configurations for each architecture. BFGS: Quasi-Newton Backpropagation; BR: Bayesian Regularization; CGF: Conjugate Gradient; GDX: Variable Learning Rate Back Propagation with Momentum; LM: Levenberg–Marquardt; NR: Resilient Backpropogation. Figure 4: Actual vs. predicted IV curves for temperature = 8.5 K, magnetic field = 300 Oe. Figures 7 and 8 show the results with respect to number of iterations (epochs) and training time, respectively. It may be observed that while the cascaded network with [5 2] neurons and _CGF_ as the training algorithm gives best results with respect to smallest number of epochs (i.e., 9), the feedforward network with [8 4] neurons and \(NR\) as the training algorithm gives the best results with respect to minimum training time (i.e., 0.109 s). These two parameters--epochs and training time--are specifically more useful in studies requiring real-time training and prediction than MSE, since smaller delays would yield faster systems. Note that each of the Figures 6-8 corresponds to a temperature of 8.5 K and a magnetic field of 300 Oe. We have summarized the best results for each measured parameter in Table 3. It can be noted that although the feedforward network with \(BR\) gives minimum MSE, it takes a large number of iterations and training time to converge. Naturally, it will not be the optimum choice in real-time systems.