## 1 Introduction  
When coal mine equipment safety resources are invested, the rated safety resources need to be reasonably allocated to each safety investment index, so as to maximize the safety economic benefits [1]. In the actual investment, most of the coal mining enterprises for the equipment safety investment of insufficient attention and lack of scientific decision-making methods, the investment method is also mainly based on experience. The uncertainty of this investment decision-making makes it difficult to accurately allocate safety investment resources. As a result, the annual safety investment resources of the enterprise cannot play a practical role, and there are problems such as waste of resources, unbalanced distribution of investment, and poor safety and economic benefits. As an important part of the 'five elements' of safety production, safety investment is an indispensable safety guarantee in the process of coal mine safety production [2]. Therefore, the study of scientific coal mine equipment safety investment optimization methods to achieve reasonable allocation of coal mine equipment safety investment and maximize the safety and economic benefits has become one of the important problems that need to be solved urgently. Problem description  
Coal mining companies often rely on historical data and past experience when deciding on equipment safety investment, and there are obvious limitations to this approach. First, the decision-making process is susceptible to the subjective judgment of the decision-maker, which may lead to bias. Second, the incompleteness or potential bias of historical data can affect the accuracy of decision-making. In addition, coal mining companies usually adopt a single investment program for equipment safety, but in actual operation, equipment faces various uncertainties and complex environmental factors, and it is difficult for a single program to effectively address these challenges. Another important issue is that it is difficult to specifically quantify the economic benefits of equipment safety investment. When making decisions on safety investments, it is often difficult for decision-makers to accurately assess the actual economic benefits of the investment. This can lead to a tendency to be overly optimistic or overly conservative in assessing safety investment options, thus affecting the accuracy of the assessment. Solution idea  
In order to solve the above problems, this paper constructs a five-dimensional model of digital twin for coal mine equipment safety investment [3; 4; 5], which is driven by twin data to achieve the optimization of equipment safety investment in coal mining enterprises. Compared with traditional methods, digital twin technology has two significant advantages: firstly, it can integrate multiple types of data to provide more comprehensive support and decision-making basis [6]; secondly, it uses real-time data to continuously update the virtual model and interact with the physical entities to quickly detect and respond to deviations in the execution of safety investment and achieve continuous optimization and feedback [7]. The introduction of digital twin technology not only makes up for the shortcomings of traditional methods in terms of being data-driven, accurate, and in real time, but also improves safety economics through modeling and simulation analysis, maximizes the efficiency of resource utilization, and thus ensures the sustainability and stability of coal mine safety production. In order to establish a scientific safety investment optimization method, this paper proposes a safety investment optimization model based on the Cobb-Douglas (C-D) production function, which is able to better fit the relationship between various indicators of safety investment and accident losses, ensuring the scientific and accurate allocation of resource investment [8].