## 1 Introduction  
In studies of heat conduction problems (forward problems), the geometric shape of the studied object, thermal conductivity, initial conditions, boundary conditions, and inner heat source are all known. The temperature field in the domain can then be directly obtained using numerical heat transfer analysis. Accordingly, if the temperature values at specific points in the system are known but one or more of the initial conditions, boundary conditions, geometry, inner heat source, or thermal conductivity are unknown, these unknowns' details can be estimated using inverse problem methods. There are inverse heat conduction problems (IHCPs) that are well known to be ill posed. The IHCP has been successfully applied in many industrial and engineering fields [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11]. Inverse geometry problems (IGPs) are considered to be inverse problems in which the geometric shapes of some parts of a studied object are unknown and must be determined based on measured temperature data in other parts of the object. In most cases, these temperatures are measured on the outer surfaces of the solid body using a set of thermocouples or a thermal image scanning system. These IGPs can be distinguished by three important aspects [12]: (1) defects in detection [13; 14], (2) shape design [15; 16; 17], and (3) identification of boundaries [18]. In this paper, we focus on the boundary identification problem. IGPs, as one branch of IHCPs, have been studied by a significant number of scholars with different optimized algorithms. The optimized algorithms, which are used to minimize the objective function, are mainly divided into gradient- and non-gradient-based optimization algorithms. Gradient-based optimization algorithms include the conjugate gradient method (CGM) [19; 17; 6; 20], the Levenberg-Marquardt method (L-MM), the steepest descent method (SDM), and the Gauss-Newton algorithm. Non-gradient-based optimization algorithms include genetic algorithms (GAs), neural network algorithms (NNAs), particle swarm optimization (PSO), the artificial bee colony algorithm (ABCA), and so forth. [PERSON] et al. [19; 20; 21] used the BEM with CGM/LM to study irregular geometrical inverse problems. The boundary identification problems they studied included frost growth on an evaporating tube and irregular geometric boundary identification with time in two-dimensional unsteady heat conduction problems. [PERSON] et al. [18] determined the geometric shape of flat plate using the CGM. [PERSON] et al. [22] used the FEM and the CGM to study the internal defects of a two-dimensional pipeline and discussed the effects of the initial guess and measurement errors on the inversion results. [PERSON] et al. [23; 24] used the BEM and the CGM to study the geometrical shapes of the inner walls of a two-dimensional cylinder. They discussed the influence on the inversion results of the inner wall shape of the initial guess, the measurement error, and the number of measured points. [PERSON] [25] solved a two-dimensional steady-state thermal boundary problem using PSO. [PERSON], [PERSON] et al. [26; 27] introduced an improved PSO algorithm that can search for more solutions simultaneously, which makes it possible to give an unbiased estimate that provides a better way to find the optimal solution in the search space. [PERSON] et al. [28] applied a hybrid CGM with SPSO to solve two-dimensional steady-state boundary shape identification, which improved the convergence speed. [PERSON] et al. [29] applied the GA to estimate the size and location of a dermatome based on the information of skin surface temperature. [PERSON] [30] applied the CGM and GA to study the inversion of two-dimensional boundary conditions and analyzed the shortcomings of the CGM and GA. The CGM is currently the most popular gradient-based optimization algorithm. However, the CGM, which is a local search algorithm, also has drawbacks, such as that it easily converges to a locally optimal solution. The inversion result depends crucially on the initial guess.