## 2 Framework for Coal Mine Equipment Safety Investment Based on Digital Twins
### Digital Twin-Based Process for Coal Mining Equipment Safety Investment  
The process of safety investment in coal mine equipment based on digital twins, as depicted in Figure 2, consists of three stages: the observation stage (Step 1), the analysis stage (Steps 2-5), and the decision stage (Step 6).  
Step 1: Digital twin modeling for equipment safety investment. Utilizing the five-dimensional digital twin modeling approach, establish a five-dimensional digital twin model for equipment safety investment.  
Step 2: Safety investment optimization model. Based on equipment safety investment tasks issued by the enterprise, upload them to \(VE\) to establish a safety investment optimization model. Generate alternative safety investment schemes simulated by the optimization model.  
Step 3: Accident loss prediction model. Utilizing the BP neural network based on the safety investment indicator structure (\(P_{V}\)) to analyze the rules (\(R_{V}\)) between equipment safety investment indicators and accident losses, establish an IPSO-BP model for predicting equipment safety investment accident losses. Predict accident losses for alternative safety investment schemes.  
Figure 2: Principles of digital twin-based optimization for coal mining equipment safety investment.  
Step 4: \(PE\) and \(VE\) Consistency Checks. Implement \(PE\) and \(VE\) consistency check based on \(R_{V}\), assessing the consistency between actual (\(Y_{i}\)) and predicted (\(\hat{Y_{i}}\)) accident losses in the equipment safety investment prediction model using the deviation metric (\(\frac{\left\|Y_{i}-\hat{Y_{i}}\right\|}{Y_{i}}\)). Deviation in this study is primarily based on predictive performance metrics (RMSE, R\({}^{2}\), etc.), providing a comprehensive assessment. If the fit accuracy is high, \(PE\) and \(VE\) are deemed consistent, and proceed to step 5. Otherwise, issues are analyzed and corrected in step 3 before remodeling.  
Step 5: Scheme validation. Validate alternative scheme accident loss predictions based on the accident loss prediction model. If the predicted accident loss (\(\hat{Y}\)) is lower than the historical accident loss (\(Y_{i}\)), the scheme validation passes; otherwise, proceed to Step 2 for data adjustment and optimization. Additionally, real-time \(PE\) data (monthly), such as \"coal mine safety transportation facilities, coal mine safety extraction facilities, coal mine safety ventilation facilities, emergency response facilities and accident losses\", will be provided to \(VE\) for continuous model updates, ensuring \(VE\) simulation accurately reflects \(PE\)'s current state. Based on simulation results, \(PE\) adjusts investment plans accordingly, completing the interaction between \(PE\) and \(VE\).  
Step 6: Decision-making on equipment safety investment optimization plan. Enterprise managers can make decisions on equipment investment plans based on the simulation results of the safety investment optimization model and accident loss prediction model.  
The entire process integrates \(PE\), \(VE\), \(DD\), \(Ss\), and \(CN\) modules, utilizing a data-driven approach for dynamic sensing, real-time analysis, autonomous decision-making, and precise optimization of equipment safety investment.