## 1 Introduction  
The inverse heat conduction problem is a typical inversion problem. It is based on the local temperature and other known parameters of a heat transfer system surface or interior that are used to inversely solve some unknown characteristic parameters of the system; examples are geometric boundary, thermal conductivity of materials, thermal boundary conditions (heat flux, surface heat transfer coefficient, temperature distribution), and historical temperature. Inverse heat conduction problems have been widely applied with great success in materials science, geology and environmental science, biological engineering, mechanical engineering, power engineering, construction engineering, aerospace engineering, metallurgical engineering, and other fields [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32]. The inversion of thermal conductivity as one branch of inverse heat conduction problems has been studied by a significant number of scholars with different optimized algorithms. The optimized algorithms mentioned are mainly divided into gradient-based and non-gradient-based optimization algorithms. The former includes the conjugate gradient method (CGM), the steepest descent method (SDM), and the Levenberg-Marquardt (LM) algorithm, while the latter includes particle swarm optimization (PSO), the genetic algorithm (GA), the artificial neural network algorithm (ANNA), and the ant colony algorithm (ACA). The CGM, SDM, and LM are currently used in more extensive inversion algorithms and have been used in inverse heat conduction problems. [PERSON] et al. studied a two-dimensional unsteady heat conduction system according to the temperature measured on the side of the target to be measured with the method of decentralized fuzzy inference [1]. [PERSON] solved the problem of inversion of the thermal conductivity of inhomogeneous materials with the complex variable differentiation method [19]. [PERSON] et al. solved the problem of inversion of thermal conductivity and specific heat capacity of materials based on the least square function and LM [20]. A highly accurate result of thermal diffusivity of composite materials, such as agar hydrogel and glycerol, has been inverted by [PERSON] with the LM method [21]. Using CGM, [PERSON] et al. inverted 1D and 2D transient thermal conductivity, which could be accurately estimated in noisy conditions [22; 23]. [PERSON] et al. verified the stability and effectiveness of thermal conductivity inversion of a two-dimensional transient heat conduction system analyzed with CGM [24]. [PERSON] et al. [32] used the recursion relations associated with Pade digital filters, efficiently solving real-time direct and inverse heat conduction problems. However, the gradient-based optimization algorithm is a form of local search algorithm, and is not without drawbacks, namely, that it easily settles on a locally optimal solution. The inversion result depends crucially on the initial guess. More importantly, the inverse problem of heat transfer is a typical ill-posed problem. When the temperature measurement information is not complete or there is a large measurement error, the inversion results obtained based on the gradient optimization method may deteriorate. In addition, when the number of inversion points is larger, the calculation of the gradient matrix is difficult and time-consuming, which directly affects the engineering application of the gradient-based optimization algorithm [2]. A non-gradient optimization algorithm is usually a global search algorithm, which has good adaptability and overcomes the difficulty of the gradient optimization algorithm easily settling on a local minimum. It does not involve the calculation of a Jacobian partial derivative matrix in the inversion process. Many scholars have used a non-gradient optimization algorithm in inverse thermal conductivity problems. [PERSON] and [PERSON] studied thermal conductivity and irregular inclusive geometry in a two-dimensional medium using PSO [27].