Generating Mathematical Expressions for Estimation of Atomic Coordinates of Carbon Nanotubes Using Genetic Programming Symbolic Regression

Abstract

The study addresses the formidable challenge of calculating atomic coordinates for carbon nanotubes (CNTs) using density functional theory (DFT), a process that can endure for days. To tackle this issue, the research leverages the Genetic Programming Symbolic Regression (GPSR) method on a publicly available dataset. The primary aim is to assess if the resulting Mathematical Equations (MEs) from GPSR can accurately estimate calculated atomic coordinates obtained through DFT. Given the numerous hyperparameters in GPSR, a Random Hyperparameter Value Search (RHVS) method is devised to pinpoint the optimal combination of hyperparameter values, maximizing estimation accuracy. Two distinct approaches are considered. The first involves applying GPSR to estimate calculated coordinates (uc, vc, wc) using all input variables (initial atomic coordinates u, v, w, and integers n, m specifying the chiral vector). The second approach applies GPSR to estimate each calculated atomic coordinate using integers n and m alongside the corresponding initial atomic coordinates. This results in the creation of six different dataset variations. The GPSR algorithm undergoes training via a 5-fold cross-validation process. The evaluation metrics include the coefficient of determination (R2), mean absolute error (MAE), root mean squared error (RMSE), and the depth and length of generated MEs. The findings from this approach demonstrate that GPSR can effectively estimate CNT atomic coordinates with high accuracy, as indicated by an impressive R2≈1.0. This study not only contributes to the advancement of accurate estimation techniques for atomic coordinates but also introduces a systematic approach for optimizing hyperparameters in GPSR, showcasing its potential for broader applications in materials science and computational chemistry.