# Neural Multivariate Grey Model and Its Applications
## 1 Introduction
The computation of ResNet's features can be regarded as the Euler method for solving an ordinary differential equation [25; 26]. Many scholars are interested in NODEs because they link differential equations with neural networks, resulting in a novel neural network domain. They are now being used in a variety of deep learning research [27; 28; 29; 30]. Based on NODE, the neural ordinary differential grey model (NODGM) [31] has been proposed as a novel grey model. The NODGM model incorporates an innovative whitening equation, allowing the learning of the prediction model through the training process. Recently, [PERSON] [32] proposed a time-delayed nonlinear neural grey model based on LSTM. However, these models are limited to a single variable and do not take into account the relevant factors, so it cannot be generalized to more realistic scenarios with multiple variables. Additionally, there is also a demand to improve the grey model's accuracy. The modeling process in GM(1, N) contains multiple variables or factors, allowing for a more comprehensive analysis of systems involving multiple influencing factors. It is an extension of GM(1, 1). This extension enables the GM(1, N) model to handle and consider the influence of multiple variables, thereby enhancing its applicability to complex systems where interactions between various factors need to be considered in prediction or analysis. To the best of my knowledge, there has not been any research on integrating NODE with GM(1, N) models. This paper aims to introduce a new model termed the neural ordinary differential multivariate grey model (NMGM). Unlike the conventional GM(1, N) model, which uses the least squares technique for parameter estimation, the NMGM model will integrate NODE methodology, offering a new modeling method for multivariate grey models. When working with a small sample size, the approach is more likely to lead to overfitting. This study presents a novel grey prediction model that utilizes NODE to establish the whitening equation, aiming to address these issues. Neural network training will reduce overfitting and enhance forecast accuracy in the model. The model does not require a defined structure, as the optimum model is derived through training and can be solved using advanced numerical methods. Therefore, our NMGM model can have better adaptability and generalization. The main contributions of this study can be summarized as follows:  
1. We propose the neural ordinary differential multivariate grey model (NMGM), which is a novel multivariate grey model that is based on the NODE. Our goal is to improve the accuracy of predicting data that include insufficient or limited information. 2. The NMGM model employs gradient descent as the training algorithm to obtain the model's parameters, as opposed to the conventional least squares method. This approach enhances the precision of the model. The optimal model parameters are obtained for this study using the adjoint sensitivity method. 3. Our model is validated in two separate cases. When predicting China's per capita energy consumption from 2012 to 2021, our model obtains a mean absolute percentage error (MAPE) of 0.2537% and 0.7381% in the test and validation sets, respectively. The MAPE of our model for predicting China's total renewable energy from 2013 to 2022 is 0.9566% in the test set and 0.7896% in the validation set. Our prediction results are more accurate than those of other models, which is extremely valuable to national energy management and policymaking. The subsequent sections of this work are organized in the following manner. In Section 2, we present the fundamental concept of the conventional GM(1, N) model and the neural ODEs. Section 3 presents our innovative NMGM model, referring to the model procedure, parameter optimization, and performance evaluation. The proposed model is utilized in Section 4 to simulate and predict China's energy consumption per capita and total renewable energy. Finally, our conclusions are presented in Section 5.