# Neural Multivariate Grey Model and Its Applications
## 5 Conclusions  
This paper introduces the neural ordinary differential multivariate grey model (NMGM) as a novel grey model for time series forecasting, which is particularly designed to tackle the challenges posed by sparse data in grey prediction. NMGM is distinguished by its utilization of neural ordinary differential equations (NODE) during training, ensuring a robust structural compatibility that significantly enhances its predictive capabilities. The NMGM model is more adaptive than prior grey models with artificially specified structures, and it can train an appropriate model from a range of sample data. Meanwhile, unlike the least squares technique, the method optimizes parameters using forward and backpropagation, preventing overfitting and improving the mode's prediction performance. As a result, the NMGM performs better in terms of generalization and accuracy than models built using the least squares method.  
Our new model outperforms four existing grey prediction models in two practical cases, affirming its superior efficacy and performance. Similar to previous grey models, our model can be used in scenarios where the data have some regularity and trends, and the amount of data is relatively small, such as an air pollution index, traffic flow, sales forecast, energy consumption, and demographics. The experimental results further highlight the ability of the NMGM model to accurately predict time series with a small number of samples. With more accurate predictions, our models can provide governments or commercial organizations with more accurate data to make better decisions.  
Our novel model enhances the accuracy of multivariate grey prediction models, contributing to the advancement of the theoretical underpinnings of grey prediction models. The advantages of this novel approach will be comprehensively explored across various real-world scenarios. In the future, we plan to promote the model's application in areas such as sales forecasting and traffic analysis. Furthermore, we will investigate greater-depth model optimization based on challenging scenarios, such as typical situations including data with time delay or periodicity.  
Conceptualization, Q.L. and X.Z.; methodology, Q.L.; software, Q.L.; validation, X.Z.; investigation, Q.L.; writing--original draft preparation, Q.L.; writing--review and editing, X.Z.; supervision, X.Z.; funding acquisition, X.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the [National Natural Science Foundation of China] grant number [62372366].  
Not applicable.  
Publicly available datasets were analyzed in this study. These data can be found here: [[https://data.stats.gov.cn/easyquery.htm/](https://data.stats.gov.cn/easyquery.htm/)]([https://data.stats.gov.cn/easyquery.htm/](https://data.stats.gov.cn/easyquery.htm/)) (accessed on 8 November 2023) and [[https://www.irena.org/Publications/2023/Mar/Renewable-capacity-statistics-2023](https://www.irena.org/Publications/2023/Mar/Renewable-capacity-statistics-2023)]([https://www.irena.org/Publications/2023/Mar/Renewable-capacity-statistics-2023](https://www.irena.org/Publications/2023/Mar/Renewable-capacity-statistics-2023)) (accessed on 9 November 2023). Due to URL permission issues, it is available by the authors on request.  
The authors would like to thank [PERSON] and [PERSON] for their invaluable contribution to project management and the validation of the results.  
The authors declare no conflicts of interest.