## 1 Introduction  
In quantitative structure-property relationship (QSPR) modeling, some mathematical and artificial intelligence methods are used to explore the chemical and physical properties of various substances. These methods, including mathematical statistics, machine learning methods, and artificial intelligence methods, can reflect the relationship between the activity and structure of compounds. Through QSPR studies, the relationship between the structure and activity of compounds can be mined [1; 2]. The QSPR model can be used to predict the activity of unknown materials and discover key influencing factors of the activity of related substances, such as groups or substituents determining the activity of the molecular structure [3; 4]. Nowadays, QSPR has been applied in the fields of computer science, chemistry, materials science, medicine science, and life sciences [5; 6]. The establishment of the QSPR model mainly involves the following steps: acquisition of experimental data, construction and optimization of the molecular structure, calculation and screening of molecular descriptors, establishment and verification of the model, etc. First of all, the variable selection is important in many fields, such as spectroscopy [7; 8], QSPR [9; 10], and other fields [11; 12]. The selection of molecular descriptors largely determines the quality of the QSPR model [13; 14; 15]. The step of molecular descriptor screening aims to reflect more structural information so that there is no noise in the descriptors. Many methods have been developed to screen molecular descriptors and can be mainly divided into two categories [16; 17; 18]. The first category includes the common methods, such as Akaike information criterion (AIC), Bayesian information criterion (BIC), and forward/backward/bi-directional stepwise multiple linear regression (MLR). The second includes the modern search algorithms, such as genetic algorithm (GA), simulated annealing algorithm (SA), ant colony algorithm (AC), particle swarm optimization (PSO), and other swarm intelligence algorithms [7; 11; 19; 20; 21]. The common methods are the most simple and efficient, but their overall performances are low in complex nonlinear problems. The modern search algorithms based on the optimization strategy have obvious advantages and can search for optimal variables and deal with complex large data points. The model establishment is important in the QSPR study and commonly used QSPR models include two-dimensional (2d), three-dimensional (3d), and four-dimensional (4d) models [22; 23; 24]. According to the modeling ideas, these methods can be divided into linear and nonlinear QSPR methods. Linear methods mainly include multiple regression methods (MLR), partial least squares (PLS), and principal component regression (PCR) [25]. Nonlinear methods include support vector regression (SVR) and artificial neural network (ANN) methods [26; 27; 28; 29; 30]. However, the QSPR study based on various artificial intelligence algorithms also has some shortcomings, such as high computational cost [31]. Therefore, it is necessary to develop a QSPR model with high accuracy, high efficiency, and good stability. The pKa value is a key parameter of some compounds, but its determination experiments are cumbersome. Therefore, it is important to develop a pKa prediction model with high accuracy. [PERSON] et al. [32] proposed a model based on ANN and PLS to predict the pKa of aromatic acids and alkyl acids. [PERSON] et al. [33] developed a model with radial basis function artificial neural network (RBF ANN) and the heuristic method (HM) and obtained the better performance in pKa prediction. These studies showed that ANN has outstanding performance in pKa prediction. However, the performance of ANN is sensitive to its parameters and training algorithm. Many artificial intelligence algorithms, including various evolutionary algorithms, are applied in ANN training.