## 1 Introduction  
Due to the particularity and complexity of geological conditions in engineering, various rock parameters in underground engineering projects present great uncertainties, which increase difficulties and variables during construction. Although some parameters can be obtained by laboratory tests or in situ tests, due to the size effect, there can be large errors in the parameters, which cannot meet the needs of practical engineering. At present, the rock parameter back-analysis method has become an important means of solving this problem, gradually forming a system of theories and methods that can be used in engineering practice [1, 2]. Underground engineering back-analysis is a method that uses displacement, stress, strain and other measured information generated by surrounding rock during underground engineering construction to establish a model and carry out numerical simulations to deduce rock parameters and initial load conditions [3, 4, 5, 6]. Displacement information is easier to monitor and obtain than stress data and other measured information, and the displacement back-analysis method is the most widely used method for back-analysis in underground engineering [7, 8]. In the displacement back-analysis process, to obtain more practical geomechanical parameters and align the calculated results with field measurement information, the rock parameters are converted to optimization parameters, and the error between the calculated results and the field measurement information is taken as the optimization objective function; this approach is called the optimization back-analysis method. After decades of research, the optimization of back-analysis methods has been continuously updated and has iteratively injected vitality into the development of the back-analysis field. Traditional optimization methods, such as the simplex method [9], conjugate gradient method [10] and semianalytical method [11], have provided impetus in the initial development of the displacement optimization back-analysis method. However, traditional optimization methods can encounter problems, such as slow convergence speed or weak adaptability, which cannot meet the needs of underground engineering back-analysis [12; 13]. Therefore, with the continuous development of the internet and computer technology, some bionic intelligent algorithms, such as the genetic algorithm (GA) [14; 15; 16], particle swarm optimization (PSO) [17; 18; 19], and ant colony optimization (ACO) [20], have been gradually introduced into geotechnical engineering and applied in the back-analysis of rock parameters. Since bionic intelligence optimization algorithms are stochastic optimization algorithms, a large number of fitness function evaluations are often required to obtain the global optimal solution for complex optimization problems. However, because the relationship between the parameters and the objective function is difficult to establish through simple formulas, it is generally necessary to use time-consuming numerical calculations to establish the implicit functional relationship, which leads to low computational efficiency and is difficult to apply to engineering practice [21]. In the optimization back-analysis, this problem can be effectively solved by replacing part of the time-consuming numerical calculation with the surrogate model. Genetic algorithm (GA) and artificial neural network (ANN) strategies have been combined to calculate the geomechanical parameters and in situ stresses [22]. When using the back-analysis method to calculate the seepage field parameters of an earth-rock dam, researchers used a support vector machine (SVM) to establish the nonlinear relationship between the seepage parameters and the water head. Then, an objective error function was used as the fitness value of the PSO algorithm, and the seepage parameters were identified by the PSO algorithm [23].