## 1 Introduction  
Underground mining is gradually becoming the primary method of extraction for mines, with increasingly prominent challenges. Compared to open-pit mining, the complex geological environment, variable mining conditions, and numerous risk factors significantly impact the safety of underground mining. Under the influence of multiple factors, rationally setting stope structural parameters is crucial for maintaining the stability of mined-out areas [1,2]. On one hand, it enhances stope stability and reduces the risk of safety accidents such as roof falls and wall collapses. On the other hand, it improves the ore recovery rates and the economic efficiency of the mine, contributing to the sustainable development of mineral resources. Therefore, the study of optimizing stope structural parameters is of great significance in the mining process [3]. Based on the research and analysis of stope structural parameter optimization both domestically and internationally, commonly used methods include the Mathew stability chart method and the FLAC3D numerical simulation method for optimizing stope structural parameters, as well as multi-objective optimization methods [4, 5, 6]. The Mathew stability chart method is primarily used to estimate the exposed area of the stope roof and the exposed area of the filling body sidewalls. This evaluation method introduces various correction factors and, based on rock mass quality grading, comprehensively considers multiple on-site influencing factors to perform a multi-criteria assessment of the actual engineering. The FLAC3D numerical simulation method [7,8] has a wide range of applications and can simulate various mechanical behaviors of three-dimensional soils, rock masses, and other materials under stress conditions. It is commonly used to optimize stope span,stope length, and stage height parameters. However, it is less effective when discussing issues such as the exposed area of the filling body sidewalls. Numerical simulation analysis enables the use of finite difference software for three-dimensional modeling of complex ore bodies and stopes, ensuring efficient analysis of complex spatial stability. However, numerical simulation analysis primarily focuses on assessing the mechanical characteristics of stopes, often overlooking economic benefits to ensure stope stability. Therefore, multi-objective optimization of stope parameters is the key to addressing this issue [9; 10; 11]. Optimizing stope structural parameters typically involves adjusting stope dimensions, such as roof thickness, pillar width, and stope span. This process usually requires analyzing the relationship between these dimensions and their mechanical responses. Multi-objective optimization methods involve using optimization-based mathematical models combined with numerical simulation software to optimize stope structural parameters. Given that stope structural parameters are influenced by multiple factors, multi-objective optimization overcomes the limitations of single-objective optimization focused solely on mechanical responses by incorporating economic factors such as cost, ensuring both economic and safe production in mining. [PERSON] [12] and others constructed a neural network model based on factors such as pillar thickness and pillar interface dimensions. They used chaotic optimization methods and ANSYS finite element numerical simulation software to determine stope structural parameters under different technical mining conditions to ensure safe ore recovery and also improve ore recovery rates. [PERSON] [13] and others used numerical analysis software to determine the safety factors for parameter combinations within a selected range. They utilized these safety factors as training samples for a BP neural network to model the nonlinear relationship between parameter combinations and safety factors. This approach aimed to find the optimal solution for the fitness function and the corresponding parameter combination.