## 6 Conclusions  
For machine learning methods, the IID assumption is a basic guarantee that the model obtained through the training data can achieve good results in the test set, and it requires a high degree of data similarity between the test set and training set. However, the complex  
Figure 12: Evaluation indicators of different prediction models: (**a**) RMSEs of different prediction models; (**b**) MAEs of different prediction models; (**c**) PEVs of different prediction models. petroleum engineering environment cannot satisfy its assumptions, which leads to the instability of the machine learning method on the test set. Based on the quantification of data similarity, this paper establishes a hybrid model combining a machine learning method and an empirical method, and it defines the AE-ED to quantify data similarity. According to the AE-ED value of each new piece of input data, the hybrid model chooses the corresponding single model to calculate. We apply the empirical model to deal with data with low data similarity, which cannot be processed stably and accurately by the machine learning model. The physical model guarantees accuracy, while the empirical model guarantees robustness and generalization. Therefore, the hybrid model combines the high prediction accuracy of the machine learning method with the good robustness and generalization of the empirical method, overcoming the shortcomings of any single model. 1. Calculate the AE-ED value of each new piece of data in the test set, and we find that the AE-ED values of many of the data exceed the threshold, which does not satisfy the IID assumption. This proves that quantifying the data similarity is very necessary;
2. Among the empirical models established in this paper, the Motahari model performs better. Meanwhile, RF is the best-performing machine learning model;
3. The hybrid model outperformed the two single models that compose it, with average reductions of 33% in the RMSE, 45% in the MRE, and 41% in the PEV. It has high prediction accuracy, as well as good robustness and generalization;
4. Because the hybrid model is a combination of the physical model and empirical model, it takes longer to run than the two single models that compose it;
5. Among the three hybrid models, the Mota-RF hybrid model has the best evaluation indicator and model performance, and it is the most suitable for the ROP prediction of the field when the drill-bit type is KM1662;
6. In the hybrid model, the main role of the empirical model is to make up for the low robustness and generalization of the machine learning model when the data similarity is low. According to the research results of this paper, the best-performing empirical model is the Motahari model, the best-performing machine learning model is the RF model, and the best-performing hybrid model is the Mota-RF hybrid model that combines them. We do not believe that simply combining the best-performing physics model with the machine learning model will result in the best hybrid model. For future work, we plan to focus on this issue. In addition, the model in this paper mainly considers the interaction between the drill bit and the rock, while ignoring the motion state of the drilling tool. Therefore, we plan to use the wellbore trajectory and the stress state of the drilling tools in the wellbore as input features to establish a more stable and accurate ROP prediction model to cope with complex working conditions in the field, which is applicable to the ROP prediction of both vertical wells and directional wells. Conceptualization, F.Z. and H.F.; methodology, F.Z., Y.L., and H.Z. ; software, Y.L. ; validation, F.Z. and R.J.; formal analysis, H.Z. and F.Z. ; investigation, H.Z. and R.J.; resources, Y.L. ; data curation, F.Z. ; writing--original draft preparation, F.Z. ; writing--review and editing, H.F.; visualization, R.J.; supervision, Y.L. ; project administration, H.F.; funding acquisition, H.F. and H.Z. All authors have read and agreed to the published version of the manuscript.