# Neural Multivariate Grey Model and Its Applications
## 1 Introduction  
Forecasting time series has consistently been of great importance for domains such as finance, economics, markets, and social issues. Various forecasting methods emerge indefinitely as science and technology advance. There are numerous models and methods available for time series, including LR (linear regression), ARIMA [1], LSTM, and neuron models [2, 3]. However, large sample sizes are required for building these prediction models. As we all know, it can be challenging to gather big samples of data for some applications in the real world. The conventional model of forecasting is ineffective when dealing with such a tiny sample set. Recently, there have been numerous methods for making predictions from small sample sizes, including neural networks [4], fuzzy theory [5], and grey system theory [6, 7]. Professor [PERSON] was considered a pioneer of grey systems theory in 1982 [8]. This theory has evolved into a valuable framework for dealing with uncertain or limited information in various fields. The theory employs mathematical tools, such as grey differential equations and grey accumulation operations, to progressively refine models and extract inherent patterns in the system. Various grey models have demonstrated remarkable predictive and analytical capabilities in numerous real-world scenarios, including in agriculture, meteorology, geology, the economy, management [6], energy [9, 10, 11], industry [12, 13], smog pollution [14, 15], and society [16, 17]. The grey model is categorized into univariate and multivariate types. The GM(1, 1) model is a univariate model that consists of a first-order difference equation. The GM(1, N) model, within the realm of multivariate modeling, comprises a primary variable denoting the system characteristic when other N-1 variables represent the influencing factors. However, the GM(1, N) model's accuracy has been impacted by its imprecise time response, which restricts its applicability. As a result, the GM(1, N) model has undergone enhancements by several researchers, yielding substantial advancements and the creation of various grey multivariate prediction models. A novel optimized grey prediction model was created by [PERSON] [18], and various experiments were conducted to illustrate the efficiency of the model. [PERSON] [19] enhanced the flexibility of the multivariable grey model by integrating time power terms into a prediction model for multivariable grey. [PERSON] et al. introduced a novel multivariable grey optimizing model called OGM(1, N) [20] to address issues related to parameter defects, mechanism defects, and structural defects. By applying dynamic background values, [PERSON] et al. [21] developed a new multivariate grey prediction model and optimized its parameters using the whale optimization method. [PERSON] et al. [22] present a novel multivariate grey model known as the GM(1, N l sin) power model, which aims to address the limitation of grey multivariate prediction models in accurately simulating systems that exhibit periodic oscillations. [PERSON] et al. expand the prediction theory of the DGM(1, N) model structure by introducing action duration, time delay, and an intensity coefficient, and proposes the ATIGM(1, N) [13] model, which is effective at forecasting time delay data. The outcomes of these studies have strengthened multivariate grey prediction models and substantially expanded the scope of grey system theory; both have resulted in extensive application. These variants of GM(1, N) have consistently demonstrated a superior performance in tasks related to forecasting time series data. Neural ordinary differential equations (NODE) [23] are a class of neural networks introduced to address continuous-depth models. These networks utilize ordinary differential equations to articulate the neural network's dynamics, enabling a flexible approach to depth rather than relying on a fixed number of layers. NODEs build on this idea by considering ResNets [24] as a specific case in which the number of layers becomes infinite.