## 1 Introduction  
Uniaxial compressive strength (UCS) is one of the most important parameters for determining the behavior of intact rocks, both in geotechnical and mining engineering [1, 2, 3, 4, 5, 6]. Initially, the UCS was obtained mainly through laboratory uniaxial compression tests using the standards proposed by the American Society for Testing and Materials (ASTM) or the International Society for Rock Mechanics (ISRM) [7, 8, 9, 10]. Nevertheless, the laboratory experiments of UCS have been controversial for three reasons, including time-consuming, cost-ineffective, and rock sample quality-dependent [2, 11, 12, 13]. Therefore, it is practically meaningful and scientifically significant to develop economical and effective but robust methods to obtain UCS to meet the needs of engineering practices and research. Several scholars used parameters to establish single regression formulas to predict UCS, e.g., the Schmidt hammer rebound number (SHR), P-wave velocity (V\({}_{\text{P}}\)), and point load strength (PLS) [11, 14, 15, 16, 17, 18, 19]. Nevertheless, these empirical relationships between a single parameter and UCS are not satisfactory in rock engineering [20, 21]. To tackle this problem, multiple regression analysis (MR) has been developed to predict UCS considering at least two parameters related to the rock properties (Table 1). It is found that the prediction accuracy for most of the multiple regression equations is not sufficient. Furthermore, some research proved that the prediction performance of artificial intelligent models is better than that of traditional statistical methods [22]. Therefore, an increasing number of research seeks to predict UCS using artificial intelligence (AI) technologies, including artificial neural network (ANN) [20; 23; 24; 25; 26], fuzzy inference system (FIS) [27; 28; 29; 30], support vector machine (SVM) [31; 32; 33; 34], random forest (RF) [35; 36], adaptive neuro-fuzzy inference system (ANFIS) [12; 37; 38; 39], multi-layer perceptron (MLP) [32; 40], gene expression programming or gene programming (GEP or GP) [13; 21; 41; 42], and genetic algorithms (GA) [43]. Compared to other AI methods, the extreme learning machine (ELM) was rarely reported to predict UCS of rock samples [44]. On the contrary, ELM has been widely used to solve a variety of engineering prediction problems such as backbreak, flyrock, and penetration rate [45; 46; 47; 48; 49; 50]. ELM was firstly developed by [51] based on a special neural network architecture, namely the single-layer feed-forward neural network (SLFN). To improve the prediction accuracy of ELM, [PERSON] et al. [52] developed a kernel-based ELM (KELM) model which is less affected by collinearity. Nevertheless, the single AI model sometimes falls into a local minimum and results in poor prediction performance [53; 54; 55]. The meta-heuristic algorithm based on biological behavior was used as an effective method to solve optimization problems [19; 56; 57; 58; 59; 60; 61; 62; 63; 64]. A novel model consisting of KELM and the grey wolf optimizer (GWO) was developed to predict UCS in this study. Besides, the other five models were also implemented to predict UCS of rock samples and compared their results with the prediction performance of the KELM-GWO model. These five models included four AI models and one empirical model, i.e., ELM, KELM (initial), deep extreme learning machine (DELM), back-propagation neural network (BPNN), and an empirical formula.