# Prediction of Critical Currents for a Diluted Square Lattice Using Artificial Neural Networks
## 1 Introduction  
Mixed state in superconductors is the sign of existence of vortices, which is the most interesting research area in low temperature physics. The vortices can be in the form of liquid, glassy, or crystalline phases. These vortex phases can be studied in high temperature superconductor systems and type II superconducting thin films with an array of dots/antidots. Over the last few decades, various properties of superconducting thin films with an array of artificial pinning centers have been explored [1; 2; 3; 4; 5]. Different geometrical structures [2; 3; 5; 6; 7; 8; 9; 10] have been used in the composition of the array of dots or antidots. Previous works [10] showed that using a diluted array of antidots increases pinning effect along with energy conservation. In this work, an experimental setup based on a diluted square array of antidots is used to measure its current-voltage (IV) behavior. It has been noted in previous works [11] that repeated transport measurements may become costly and cumbersome to obtain. As a result, there is a dire need to come up with a theoretical or a formal model that could approximate the IV curves of superconducting films, and thereby relieve the researchers from repeatedly measuring these physical properties. Artificial neural networks (ANNs) are among the most widely adopted techniques for modeling complex systems. The concept of ANN is derived from the actual working of a human neuron system [12], in which different neurons connect with each other through some network's coefficients (called weights). The working of ANNs is well addressed in literature [13]. ANNs learn and identify relationships between systems' parameters: this gives them an astonishing approximation or prediction capability. Other advantages associated withthe use of ANNs include modeling non-linearity [14; 15; 16; 17], fault tolerance, parallelism, robustness of the learning process, and ability to handle fuzzy information [18; 19]. The advantage of using ANNs over other statistical methods--such as linear and nonlinear regression techniques--has been advocated on multiple occasions for various applications. For example, [20] made a comparison between the two techniques for the prediction of yarn tensile properties, [21] carried out the same comparison for Iran's annual electricity load, and [22] compared ANN with linear regression models for predicting hourly and daily diffuse fraction. All of them concluded that ANN was the better prediction approach. Despite their tremendous features, ANNs have had a very limited application in the field of materials science [11; 23; 24]--let alone the prediction of IV curves. In [11], ANNs were used for the prediction of the current-voltage curves for a square array of nano-engineered periodic antidots. Diluted square arrays--which are formed by removing a quarter of the sites from the original square lattice--offer a larger interstitial area in comparison to the original square lattice. This means that a large number of interstitial vortices may easily be accommodated, leading to increased energy conservation; this is commonly referred to as _caging effect_[3]. In this work, we predict the IV curves for a diluted square array of antidots, and propose a framework based on ANN for extrapolating the IV behavior for a wide range of temperature and magnetic field values. In addition to this, we present a thorough comparison of three different ANN architectures trained with six different training algorithms for the framework. The comparison of training algorithms is based on prediction accuracy in terms of mean squared error (MSE), number of iterations needed to converge (epochs), and training time. Our findings may be used as a benchmark in any followup work concerning the study of IV characteristics of any regular or diluted lattice, since we pinpoint the pros and cons of several architectures and training algorithms, and conclude on the most suitable options for our specific application. The rest of the paper is organized as follows.