Synergistic Evolution of China’s Green Economy and Digital Economy Based on LSTM-GM and Grey Absolute Correlation

Abstract

This study aims to understand the synergistic evolution of the green and digital economies towards sustainable development. Previous research lacked quantitative analysis, which hindered the development of a comprehensive understanding. An evaluation index system is established using the CRITIC and entropy weight combination methods. The TOPSIS model is utilized to evaluate indicators and derive a comprehensive development index for both economies. The LSTM-GM model is employed to predict the evolutionary trends for the next five years. The absolute grey correlation model is applied to analyze historical and future synergistic evolutionary trends. Findings show increasing levels of green and digital economic development. The digital economy promotes green economic development by enhancing efficiency through innovation and upgrades. The green economy facilitates the structural adjustment of the digital economy by reducing emissions and enhancing resource utilization. Predictions indicate a steady growth in both economies and an increasing synergistic evolution. Based on the analysis, policy recommendations are proposed to promote the integration and development of the digital and green economies, facilitating high-quality synergistic growth.

1. Introduction

Since the reform, the rapid development of China’s economy has driven the progress of industrial intelligence, urban digitization, and industrial greening. This has, in turn, promoted the development of the digital economy and improved resource utilization and the ecological environment to some extent [1,2,3].

As nations worldwide strive to build green and efficient economic development models, China’s green development index has increased by 77.5%, 12.2%, and 27.9% in the areas of sustainable development, resource utilization, and environmental protection, respectively. Meanwhile, the global digital economy has exceeded USD 38.1 trillion, with an annual growth rate of 15.6%, accounting for 45.0% of GDP [4]. Notably, the United States and China collectively represent approximately 90% of the world’s largest digital platform market value, as highlighted by the United Nations Conference on Trade and Development. The expansion of the digital economy provides new pathways for innovation and entrepreneurship in emerging fields such as digital currency, finance, self-publishing, and e-commerce [5].

In the integrated development of the digital and green economies, various challenges arise due to the emergence of new industrial structures, technologies, energy sources, and business models [6]. The previous studies primarily focused on analyzing the digital economy’s impact on the efficiency of green development, but there has been less research on the law of synergistic evolution between the green economy and the digital economy [7,8]. Therefore, it is crucial to effectively reveal the law of synergistic evolution between the green economy and the digital economy in order to promote the rapid and sustainable development [9,10,11].

Considering these factors, our research endeavors to shed light on the intricate dynamics between the digital economy and the green economy within the context of China. In our research, we meticulously curated a comprehensive indicator system comprising 41 green economy (GE) indicators and 23 digital economy (DE) indicators. This selection was not arbitrary but was made with careful consideration of several critical factors. Our choice of indicators was grounded in reputable and authoritative reports, including the “2022 China Digital Economy Development Index Report” and the “14th Five-Year Plan for Digital Economy Development in China.” These reports provide a solid foundation for selecting indicators that are not only up to date but also reflective of the current landscape of China’s digital and green economies. We meticulously assessed their relevance to the study’s core focus, ensuring that each selected indicator could effectively capture essential aspects of both the green and digital economies. This alignment was crucial to enable a comprehensive examination of the intricate interplay between these two vital dimensions of economic development. Finally, we consulted the work of various scholars who have previously developed frameworks to assess green economies. By incorporating a wide array of indicators, we aimed to provide a nuanced and in-depth analysis of the complexities and subtle distinctions within both economic domains. This expansion of the number of indicators was undertaken to facilitate a thorough exploration of the multifaceted nature of these economies.

In summary, based on the selected 41 green economy indicators and 23 digital economy indicators, combined with an authoritative department’s statistical data, this paper established a comprehensive indicator system for the green and digital economies. We applied the TOPSIS method to evaluate the integrated development indices for both economic subsystems from 2008 to 2022. In addition, considering the limitations of the grey prediction model, the LSTM model was used to optimize the results of the grey prediction model and predict the future development trend of the green and digital economies. Finally, by comparing the law of synergistic evolution in history and for the future, we offer specific policy suggestions to foster the integrated development of the green and digital economies.

2. Literature Review

This paper examines the fields of the green and digital economies, making it necessary to sort out the previous research from the perspective of the development status of both economies and their synergistic relationship.

2.1. Green Economy

Carson first proposed the concept of green development in his book Silent Spring (1963) to address the ecological issues caused by industrial pollution [12]. Later, economists such as Pearce coined the term “green economy” to emphasize its social and ecological dimensions [13]. In its 2022 report, the United Nations Environment Programme defines a green economy as a low-carbon, resource-efficient, and social economy [14]. The term “green economy” is also used to describe the effective use of resources in the consumption process [15]. In addition, numerous scholars have developed comprehensive green economic development indices, and their findings consistently indicate the overall growth of the green economy [16,17,18]. This growth can be attributed to the increased awareness and adoption of new green energy, resulting in improvements in the ecological environment. One study constructed a green productivity index based on the global Malmquist–Luenberger productivity index to evaluate the development of the low-carbon economy in China. A panel vector auto-regression model was adopted to further analyze the endogenous interactions and dynamic relationships among aspects of the green economy and its influencing factors [19]. Another study aimed to examine the potential for green economy measures to create green jobs in the agriculture sector [20]. Questions have arisen as to whether the OFDI of China brings green spillovers back home, and whether the government should encourage OFDI to further promote the domestic green economy. To answer these questions, researchers have examined empirically the reverse spillovers of China’s OFDI on its domestic green development [21]. One study investigated how increasing economic development affects the green economy in terms of CO2 emissions, using data from 44 countries in sub-Saharan Africa for the period 2000–2012 [22]. For the case of 278 Chinese prefecture cities, the data envelopment analysis game cross-efficiency model was used to measure the GEE between 2003 and 2017 [23]. Another study employed the well-known framework for strategic sustainable development (the natural step framework) to comparatively identify the relative and integrated contributions of the three narratives for global net sustainability [24]. Other influential work includes [25,26,27,28].

2.2. Digital Economy

The digital economy has experienced significant growth in various industries [29,30]. Nevertheless, the rapid expansion of the digital economy has also brought challenges, such as the digital divide, digital security, and uneven ecological development. The United Nations Conference on Trade and Development (UNCTAD) has highlighted the massive imbalance caused by the data-driven digital economy [5], while the International Monetary Fund (IMF) has emphasized the importance of cybersecurity and resource efficiency to minimize environmental pollution [31]. There is a consensus that digital economy development is generally growing, accompanied by uneven development, inconsistent growth rates, potential crises, etc. [32,33,34]. Some studies have shown that the development of the digital economy accelerates rapidly before gradually slowing down [34]. A few scholars argue that the green economy, an environmentally friendly economic model, can solve the development imbalance caused by the data-driven digital economy [33].

Recent studies have further delved into the complexities of the digital economy. Luna investigates how digital transformation affects the capacity for social innovation, highlighting the multifaceted nature of the digital economy’s impact [35]. Meanwhile, Llop examines the welfare effects of taxing polluting export goods, shedding light on the economic and environmental dimensions of digitalization [36]. Jürgensmeier illustrates these concepts through empirical studies, emphasizing the practical implications of digital economy development [37]. Additionally, Gokhberg provides insights into measuring expenditures for digital economy development, offering a quantitative perspective on its growth [38]. Petrov assess the effectiveness of investments in digitalization within the context of a national program, emphasizing the link between digital initiatives and economic performance [39].

These contributions align with the three-scope approach introduced by Bukht and Heeks, framing our understanding of the digital economy [40]. Furthermore, Baskakova analyzes trust dynamics in Russian society, an essential factor in the digital economy’s success [41]. Chinoracky proposes a measure for assessing the scale and potential of the digital economy [42], while Borowiecki examines digital convergence in the European Union. Collectively, these studies underscore the complexity of the digital economy’s impact on society, economics, and the environment [43].

The digital economy’s rapid growth has led to challenges like the digital divide and ecological imbalances. The UNCTAD highlights data-driven disparities, while the IMF emphasizes cybersecurity and eco-efficiency. While the digital economy generally grows, some studies suggest eventual slowdowns. Scholars have explored how digital transformation affects social innovation, taxing polluting exports, and measuring digital economy expenditures. These contributions add depth to our understanding of the digital economy’s multifaceted impact. Despite the strategic importance of both, a contradiction exists between digital and green development.

2.3. The Relationship between Green Economy and Digital Economy

Research on the green economy and the digital economy has focused on four areas (Economic impacts and sustainable development, Environment and Green Innovation, Digitalization and Policy, Evaluation and Measurement) further subdivided into 10 sub-points. See Appendix A for more details:

• The impact of the digital economy on green productivity and carbon emissions: Researchers have studied the effects of the digital economy on China’s green total factor productivity (GTFP) or green total factor efficiency (GTFEE) and carbon emissions. The results show that the digital economy significantly impacts green productivity and carbon emissions in China. However, the direction and magnitude of the impact depend on the development stage of the digital economy [18,32,44].
• The impact of the digital economy on green innovation: Researchers have utilized patent data and other sources of green innovation data to examine the direct and indirect effects. These studies show that the digital economy affects green innovation through various channels, such as economic openness, industrial structure, and market potential [45,46].
• The impact of the digital economy on green growth: Studies have shown that the digital economy has a positive impact on green growth in China. However, this impact is not uniform or consistent. Studies also show that the digital economy affects green growth through various factors, such as human capital, industrial structure upgrading, energy consumption, environmental pollution, and economic growth [47,48,49].
• The impact of the digital economy on the circular economy: Studies have shown that the digital economy has the potential to influence China’s circular economy, but this impact has not been empirically or quantitatively measured [50,51,52].
• Impact of the digital economy on sustainability: Several studies highlight the profound impact of the digital economy on sustainability. Savchenko introduces a typology of sustainable development goals (SDGs) that provides a framework for sustainable urban development via digital solutions [53]. Alenkova delves into eco-innovations in production enterprises, demonstrating how the digital economy fosters innovative green practices [54]. Piao emphasizes the D2D effect, showcasing the synergistic benefits of addressing cross-sectoral sustainability challenges through digital solutions [55]. Nurova argues that the digital economy’s reduced energy consumption inherently contributes to sustainability [56]. These findings collectively underscore the transformative potential of the digital economy in promoting sustainability.
• Urban sustainability and the digital economy: Delitheou highlights cities’ pivotal role in enhancing citizens’ lives through technology and green growth [57]. The study emphasizes the importance of urban areas in bridging the gap between the digital and green economies. Xing discusses regional strategies for accelerating digital economy development while promoting innovation and green growth, showcasing the regional dimension of this relationship [58]. This research illuminates the urban context as a nexus for digital and green economy interactions.
• Economic transformations and sustainability: Ciuriak examines the global economic impact of the pandemic and its repercussions for traditional and digital trade, underscoring the evolving landscape of sustainability in a rapidly changing economic environment [59]. These studies underscore the dynamic nature of sustainability within evolving economic frameworks.
• Knowledge spillovers, innovation, and sustainability: Aldieri and colleagues offer valuable insights into the role of knowledge spillovers, innovation, and environmental considerations in achieving sustainability goals. Their work demonstrates the interconnectedness of knowledge sharing and sustainable development [60,61,62,63,64,65].
• Environmental considerations in trade and industry: Llop analyzes the welfare effects of taxing polluting export goods, providing insights into the trade-offs between economic growth and environmental sustainability [36]. Zafar investigates how biomass energy consumption impacts environmental quality, emphasizing education and technology as key factors [66]. These studies contribute to our understanding of the economic–environmental dynamics in specific sectors and regions.
• Carbon emissions and economic interactions: Xu explores spillover effects and correlations between carbon emissions and stock markets [67], while Zhen examines variations in carbon emissions in China’s exports [68]. These studies offer insights into the complex relationships between carbon emissions and economic activities in a carbon-intensive context.

Other studies have evaluated the influence of the digital economy on the development of the green economy. The findings consistently suggest that the digital economy positively contributes to the green economy. However, the relationship between the two economies is nonlinear, and the development of the digital economy can have both catalytic and potentially inhibitory effects on the green economy [3,34,69,70]. The European Green Digital Alliance argues that a certain level of digital intervention can maximize the benefits of green sustainability in the economy.

In summary, prior research has primarily concentrated on examining the influence of the digital economy on diverse aspects like green productivity, carbon emissions, green innovation, green growth, and circular economy. However, there has been a notable gap in comprehensively studying the synergistic relationship between these two economic subsystems. To bridge this gap, this paper introduces a novel green evaluation index system that employs the TOPSIS composite index model and the grey absolute correlation model to evaluate the developmental status and synergy of the digital and green economies. Furthermore, this study incorporates the LSTM neural network to enhance the predictive accuracy of the GM model and ultimately scrutinizes historical and prospective synergies. The research summarized here encompasses a broad spectrum of topics within the nexus of the green economy and the digital economy. These investigations illuminate their multifaceted interactions and their far-reaching implications for sustainability, innovation, and economic advancement. Collectively, these studies underscore the potential of digital technologies as transformative tools in advancing sustainability objectives across various domains.

3. Model Construction

This paper aims to investigate the synergistic evolution of the digital economy and the green economy. To achieve this, a comprehensive index system is constructed for the two economic subsystems, and the overall development levels are quantified using the TOPSIS comprehensive evaluation method. The synergistic relationship is analyzed using the absolute grey correlation model, and the grey prediction model is optimized using the LSTM model to predict future development trends. Comparative analyses of historical and future synergies are conducted to explore the evolutionary trends of synergy between the two economic subsystems. These analyses promote the synergy between the digital economy and the green economy in China. The research process is presented in detail in Figure 1.

3.1. TOPSIS Comprehensive Evaluation Model

This paper constructs a comprehensive indicator model using the TOPSIS method to objectively evaluate the integrated development indicators of the green economy and the digital economy. The modeling process involves the following steps:

Indicator Empowerment: The entropy weight method is used to calculate the information entropy and redundancy of each indicator, reflecting their importance. The CRITIC method assigns weights based on contrast intensity and conflict, with smaller conflicts resulting in lower weights. The combined assignment method is also employed to address contradictions and improve differentiation in the evaluation results of the single assignment method.

(1) The combined assignment method has also been widely used in various fields in recent years. It can solve the problems of contradiction and poor differentiation in the evaluation results of the single assignment method [71,72,73,74]. The steps are as follows:

(a) The calculation of information entropy plays a fundamental role in various fields, including information theory, statistics, communication systems, data analysis, and more [75,76]. The information entropy is calculated, where $p_{ij}$ is the standardized value of the ith sample of the jth indicator, n is the number of samples, m is the number of indicators, and $d_j$ indicates the amount of valid information provided by the indicator.

$$e_j=-\frac{1}{ln \left(n\right)}{\sum }_{i=1}^n\left[p_{ij}\ast ln \left(p_{ij}\right)\right]$$

(1)

$$d_j=1-e_j$$

(2)

(b) The entropy weight of each indicator is calculated. Based on the calculated entropy e(G) and e(D), the information utility values d(G) and d(D) of each indicator are calculated to obtain the weights, i.e., WG-EWM and WD-EWM [77].

$$w_j=\frac{d_j}{\sum _{j=1}^m{d}_j}$$

(3)

(c) The standard deviation is calculated to represent the comparative strength $S_j$. The correlation coefficient represents the conflict $R_j$ between indicators, and the stronger the correlation, the smaller the conflict. $X_{ij}$ is a standardized post dataset. ${\mathrm{r}}_{\mathrm{i}\mathrm{j}}$ represents the correlation coefficient between evaluation indicators i and j. This paper uses the Pearson correlation coefficient, and finally obtains the comparative strength and conflict between the green and digital economies as SG, SD, RG, and RD, respectively.

$$\left\{\begin{array}{l}{\bar{x}}_j=\frac{1}{n}\sum _{i=1}^n{x}_{ij}\\ S_j=\sqrt{\frac{\sum _{l=1}^n{\left(x_j-{\overline{x}}_j\right)}^2}{n-1}}\end{array}\right.$$

(4)

$$R_j={\sum }_{i=1,j=1}^n\left(1-r_{ij}\right)$$

(5)

(d) The amount of information C_i is calculated; the greater the amount of information, the greater the role of the jth indicator in the whole evaluation indicator system. It should be assigned more weight; the results are recorded as CG, CD.

$$C_i=S_j\times R_j$$

(6)

(e) The CRITIC objective weighting is represented by ${\mathrm{W}}_i$; the results are recorded as WG-CRITIC and WD-CRITIC.

$$W_i=\frac{C_i}{\sum _{j=1}^n{C}_i}$$

(7)

(f) The integrated weights are calculated according to formula (8), where $\epsilon$ is the coefficient, generally taken as 0.5. Finally, the integrated weights WG and WD are obtained.

$$W_X=\epsilon \times \left(W_{X-CRITIC}+W_{X-EWM}\right)\left(X\in \left[G,D\right]\right)$$

(8)

(g) To calculate the composite value, according to formula (9), multiply the composite weight matrix WX with the original sequence Gij and Dij to obtain the composite level values ZG and ZD.

$$Z_X=W_X\times X_{ij}\left(X\in \left[G,D\right],i,j\in \left[\mathrm{1,2},3\dots ,n\right]\right)$$

(9)

(2) The technique for order of preference by similarity to ideal solution (TOPSIS) is a multicriteria decision analysis method for evaluating and selecting the best decision solution. It is popular in various fields, such as engineering, management, and social sciences, due to its simplicity and effectiveness [78]. This method effectively solves the problem of dynamically adjusting indicators in this paper. Therefore, TOPSIS is a suitable method for use in this paper [79]. The main steps are as follows:

(a) Normalize the matrix. Among them, Z is obtained by multiplying the weight matrix ${\omega }_j$ and the normalization matrix $P_{ij}$ to obtain ZGi and ZDk (i and k are the number of first-level indicators).

$$Z={\left(z_{ij}\right)}_{n\times m}=\left(P_{ij}\times {\omega }_j\right)$$

(10)

(b) $z_j^{+}$ and $z_j^{-}$ represent the positive and negative ideal solutions, indicating that each indicator reaches the best and worst value in the sample, denoted as ZGj+, ZGj−, ZDj+, and ZDJ. Then, the sum of the distances between positive ideal solutions $D_i^{+}$ and negative ideal solutions $D_i^{-}$, $z_{ij}$, is calculated and becomes the current indicator value. The positive and negative ideal distances for the green and digital economies are denoted as DGi+, DGi−, DDi+, and DDi−.

$$D_i^{+}=\sqrt{{\sum }_{j=1}^m{\left(z_{ij}-z_j^{+}\right)}^2},(i=1,\dots ,n)$$

(11)

$$D_i^{-}=\sqrt{{\sum }_{j=1}^m{\left(z_{ij}-z_j^{-}\right)}^2},(i=1,\dots ,n)$$

(12)

(c) The proximity of the optimal solution $C_i$ is calculated. The range of values is [0, 1], and the closer it is to 1, the better the solution. The relative proximity between the green economy CGi and the digital economy CDi is calculated from 2008 to 2022 per year. The results are ranked according to the magnitude of their relative proximity $C_i$ to obtain a comprehensive evaluation.

$$C_i=\frac{D_i^{-}}{D_i^{+}+D_i^{-}}$$

(13)

3.2. LSTM-GM Prediction Model

LSTM was evaluated against the time-tested ARIMA model in financial and economic time series forecasting, and LSTM emerged as the superior choice, offering remarkable reductions in error rates [80,81]. In sustainable finance, Wang introduced a novel hybrid model, CEEMDAN-LSTM, for accurately predicting the green bond market, shedding light on correlations with the crude oil and green stock markets [82]. Researchers have noted LSTM’s potential in technology forecasting, aiding businesses and researchers in identifying emerging technologies [83]. These studies collectively underscored the transformative potential of LSTM and deep learning, offering actionable insights across economics, finance, sustainable finance, and technology forecasting. LSTM’s adaptability and accuracy make it a powerful tool for tackling complex real-world challenges in these fields. Predicting the development of the digital economy and the green economy has an important guiding value for policy making and industrial planning. So, in this paper, the results of the optimized grey prediction model using the long short-term memory (LSTM) model are used to predict the development trend of the digital and green economies. The GM (1, 1) model is established to predict the values of each index in the next 5 years by using a comprehensive evaluation index of the green and digital economy. The simulation error is predicted according to the grey prediction model, and the prediction results of the GM are optimized by using LSTM. The model construction steps are as follows:

(1) Grey prediction model. The GM (m, n) is a prediction method developed based on the grey system theory, where m is the order of the differential equation and n is the number of variables. This method works best when the data are incomplete, ambiguous, or uncertain [84,85,86]. The main steps are as follows:

(a) Firstly, using the comprehensive evaluation level of the green and digital economies, the 1-AGO generation of the original sequence is calculated. After obtaining the evaluation index sequence $x^{\left(0\right)}$ and generating $x^{\left(1\right)}$ after accumulation, it is represented as XG(0), XG(1), XD(0), and XD(1), respectively.

$$x^{\left(0\right)}:\left\{x^{\left(0\right)}\left(1\right),x^{\left(0\right)}\left(2\right),x^{\left(0\right)}\left(3\right),\dots ,x^{\left(0\right)}\left(n\right)\right\}$$

(14)

$$x^{\left(1\right)}:\left\{x^{\left(1\right)}\left(1\right),x^{\left(1\right)}\left(2\right),x^{\left(1\right)}\left(3\right),\dots ,x^{\left(1\right)}\left(n\right)\right\}$$

(15)

(b) The matrix $B$ and $\mathrm{Y}$ is constructed.

$$B=\left[\begin{array}{cc}-\frac{1}{2}\left[x^{\left(1\right)}\left(1\right)+x^{\left(1\right)}\left(2\right)\right]& 1\\ -\frac{1}{2}\left[x^{\left(1\right)}\left(2\right)+x^{\left(1\right)}\left(3\right)\right]& 1\\ \cdots & \cdots \\ -\frac{1}{2}\left[x^{\left(1\right)}\left(n\right)+x^{\left(1\right)}\left(n\right)\right]& 1\end{array}\right],Y=\left[\begin{array}{c}{x}^{\left(0\right)}\left(2\right)\\ x^{\left(0\right)}\left(3\right)\\ \cdots \\ x^{\left(0\right)}\left(n\right)\end{array}\right]$$

(16)

(c) The parameters $\widehat{u}$ are calculated.

$$\widehat{u}={\left(B^{T}B\right)}^{-1}{B}^{T}Y$$

(17)

(d) The development coefficient a and the amount of action b are calculated. The cumulative sequence $x^1\left(k\right)$ background greyscale, using the initial sequence $x^0\left(k\right)$ greyscale derivative, is added into Equation (18) to solve the parameter greyscale development coefficient a and the amount of action b.

$$\frac{d{x}^1\left(k\right)}{dk}+a{x}^1\left(k\right)=b$$

(18)

(e) The predicted values ${\widehat{x}}^{\left(0\right)}(k+1)$ are calculated. The parameters a and b are brought into Equation (19) to calculate the sum of predicted values for the green and digital economy.

$${\widehat{x}}^{\left(0\right)}(k+1)={\widehat{x}}^{\left(1\right)}(k+1)-{\widehat{x}}^{\left(1\right)}\left(k\right)$$

(19)

(2) Long- and short-term transient memory neural network prediction. The LSTM model can handle time series prediction well and solve the problems of gradient disappearance and gradient explosion that exist during the long series training [87]. Since the grey prediction model is weak in capturing the complexity of real-world situations, and it produced certain prediction results with high relative simulation errors in the results of this study, this paper optimizes the grey prediction model using the LSTM model to effectively capture the time series features in historical data in order to improve the accuracy of the prediction results and to solve the problem of low prediction volatility [82]. The main steps are as follows:

(a) Build the LSTM model. The indicator data are divided into training sets, test sets, and validation sets according to a ratio of 10:3:2. The original sequence is normalized as an input, the hidden layer is set to two layers in the LSTM model, and 40 neurons are set in each hidden layer.

(b) When the input is set at $i_t$, forgetting $f_t$, and output gates $o_t$, $\mathrm{w}$ is the weight matrix, $h_{t-1}$ is the output state of the previous moment, $x_t$ is the input of the current moment, and $b_c$ is the bias. The σ activation function uses a sigmoid function.

$$i_t=\sigma \left(W_i\cdot \left[h_{t-1},x_t\right]+b_i\right)$$

(20)

$$f_t=\sigma \left(W_f\cdot \left[h_{t-1},x_t\right]+b_f\right)$$

(21)

$$o_t=\sigma \left(W_o\cdot \left[h_{t-1},x_t\right]+b_o\right)$$

(22)

(c) The memory cell is set, where $\tilde{C}t$ is the new candidate value and $\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}$ is the activation function.

$$\tilde{C}t=tanh \left(W_C\cdot \left[h_{t-1},x_t\right]+b_c\right)$$

(23)

(d) The memory cell state is updated, where $C_t$ is the memory cell state.

(e) The model is tested for overfitting with the data from the validation set, and the prediction result sequences G-Lt and D-Lt are output for the green economy and digital economy after meeting the requirements.

$$C_t=f_t{C}_{t-1}+i_t{\tilde{C}}_t$$

(24)

(f) The output value is updated, where $L_t$ is the output value.

$$L_t=o_t\ast tanh \left(C_t\right)$$

(25)

(g) Through continuous iterations, the number of model layers and the number of neurons in the hidden layer are adjusted, and the weight matrix W of the model and the coefficient matrix L are output when the value of the loss function reaches the requirement.

(h) The simulation error $k$ is calculated from the results of the grey prediction model and set as the optimization coefficient. ${\widehat{X}}^{\left(0\right)}$ and ${\widehat{L}}^{\left(0\right)}$ are the prediction sequences obtained from the grey prediction model and the LSTM prediction model, respectively, and R is the final prediction result sequence.

$$R=\left(1-k\right)\ast {\widehat{X}}^{\left(0\right)}+k{\widehat{L}}^{\left(0\right)}$$

(26)

3.3. Grey Absolute Correlation Model

Grey correlation analysis is a decision-making method for evaluating and ranking alternatives based on multiple criteria. Based on the concept of the grey system theory, it is able to deal with situations where data are incomplete, ambiguous, or uncertain [88,89]. This paper focuses on the study of the degree of synergy between two economic subsystems. The grey absolute correlation model can assess the synergy between the two datasets without considering their relative positions and is more sensitive to outliers, which can well reflect the problems existing between the data. Therefore, the grey absolute correlation model was chosen in this paper to explore the synergy between the green economy and the digital economy [84,90,91]. According to the theoretical mechanism of the grey absolute correlation model, the composite indices CGi and CDi (i = 1,2, 3, …, n), n is the number of years studied, and the grey absolute correlation model is constructed as follows:

(a) $x_0^{\left(0\right)}$ is set as the reference sequence and $x^{\left(1\right)}$ as the comparison sequence, the normalized matrix X is multiplied by the weighting matrix W, and finally, the initial sequence Y0 and the comparison sequence Y1 are obtained.

$$Y_0:\left\{x_0^{\left(0\right)}w\left(k\right),k=\mathrm{1,2},\cdots n\right\}$$

(27)

$$Y_1:\left\{x^{\left(1\right)}w\left(k\right),k=\mathrm{1,2},\cdots n\right\}$$

(28)

(b) The green economy cumulative series aG(0) and the digital economy cumulative series aD(1) are calculated, where k = 1, 2,⋯, n − 1.

$$a_0^{\left(0\right)}\left(y_0^{\left(0\right)}(k+1)\right)=y_0^{\left(0\right)}(k+1)-y_0^{\left(0\right)}\left(k\right)$$

(29)

$$a^{\left(1\right)}\left(y^{\left(0\right)}(k+1)\right)=y^{\left(0\right)}(k+1)-y^{\left(0\right)}\left(k\right)$$

(30)

(c) The degree of correlation is calculated, using Equation (31), and finally the annual synergy RGD is calculated between the green economy and the digital economy.

$$R\left(\xi \left(k+1\right)\right)=\frac{1}{1+\left|a_0^{\left(0\right)}\left(y_0^{\left(0\right)}\left(k+1\right)\right)-{\alpha }^{\left(1\right)}\left(y^{\left(0\right)}\left(k+1\right)\right)\right|},k=\mathrm{1,2},\cdots n-1$$

(31)

(d) Then, using the same steps, together with the prediction results, the synergy REGD of the predicted series is calculated.

(e) The combined synergy is calculated, and the historical RGD and future synergy REGD are compared and analyzed.

$$r=\frac{1}{n-1}{\sum }_{k=2}^n\xi \left(k\right)$$

(32)

4. Empirical Research

For this paper, we used 2008–2022 data from the China Finance Yearbook, China Statistical Yearbook, China Science and Technology Statistical Yearbook, China Statistical Abstract Ministry of Finance, China Population and Employment Statistical Yearbook, Statistical Bulletin of National Economic and Social Development of the People’s Republic of China, China Education Statistical Yearbook, China Health Statistical Yearbook, World Bank, Ministry of Ecology and Environment, China Environment Statistical Yearbook, IRENA, and other official authoritative statistics. With reference to previous scholars’ studies, some trade-offs and additions were made in the selection of indicators, and finally, 49 green economy indicators and 23 digital economy indicators were constructed, for which some of the missing data were obtained by linear interpolation [92,93,94,95,96].

4.1. Indicator Construction

The construction of the green economy index system considered evaluation indices related to sustainable development, social development, economic development, resource utilization rate, waste recycling rate, social welfare, new energy, pollutant emissions, and government policy. The digital economy indicators were based on reports such as the “2022 China Digital Economy Development Index Report” and China’s “14th Five-Year Plan” digital economy development report, encompassing digital infrastructure capacity building, digital industrialization, digitalization of industry, digitalization of governance, and digital ecosystem construction.

4.1.1. Digital Economy Indicators

The digital economy indicators were categorized into five aspects, resulting in the selection of 23 indicators. Refer to Figure 2 for detailed information.

4.1.2. Green Economy Indicators

The green economy was divided into nine categories, and 49 indicators suitable for China’s green economy development were selected based on previous studies and considering data availability. Linear interpolation was used to fill in missing data. Refer to Figure 3 for a detailed view of the green economy indicator system.

4.1.3. Comprehensive Index

Based on the above TOPSIS model construction content, the entropy values were obtained according to Equation (3), the CRITIC values were obtained according to Equation (7), and comprehensive weights were obtained according to Equation (8), all of which can be seen in detail in

Table 1. Green economy composite weighting.

Indicators	CRITIC	Entropy	Combined Weights
G11	0.0647	0.0625	6.36%
G12	0.0335	0.0539	4.37%
G13	0.0865	0.0468	6.67%
G14	0.0298	0.0507	4.03%
G21	0.0694	0.0454	5.74%
G22	0.0659	0.0361	5.10%
G23	0.0334	0.0486	4.10%
G24	0.0294	0.0517	4.06%
G31	0.0299	0.0288	2.94%
G32	0.0619	0.0400	5.10%
G33	0.0652	0.0291	4.72%
G34	0.0436	0.0495	4.66%
G35	0.0395	0.0289	3.42%
G36	0.0416	0.0402	4.09%
G37	0.0858	0.0711	7.85%
G38	0.0326	0.0517	4.22%
G41	0.0323	0.052	4.22%
G42	0.028	0.043	3.55%
G43	0.0393	0.0427	4.10%
G44	0.0332	0.0487	4.10%
G45	0.0288	0.0431	3.60%
G46	0.0259	0.0351	3.05%
G51	0.0300	0.0439	3.70%
G52	0.0356	0.0567	4.62%
G61	0.0315	0.0553	4.34%
G62	0.0375	0.0621	4.98%
G63	0.0315	0.0581	4.48%
G64	0.0526	0.0392	4.59%
G65	0.0324	0.0508	4.16%
G71	0.0664	0.0283	4.74%
G72	0.0971	0.0333	6.52%
G73	0.0652	0.0827	7.40%
G74	0.0938	0.0275	6.07%
G75	0.0426	0.0245	3.36%
G76	0.0438	0.0654	5.46%
G77	0.0605	0.0735	6.70%
G81	0.0523	0.0509	5.16%
G82	0.0408	0.0516	4.62%
G83	0.0800	0.0544	6.72%
G84	0.0335	0.0506	4.21%
G85	0.0727	0.0914	8.21%
G91	0.1527	0.0618	10.73%
G92	0.1264	0.1053	11.59%
G93	0.1100	0.0991	10.46%
G94	0.1217	0.2786	20.02%
G95	0.1532	0.0584	10.58%
G96	0.1171	0.2396	17.84%
G97	0.1191	0.1059	11.25%
G98	0.0998	0.0513	7.56%

and

Table 2. Integrated weighting of the digital economy.

Indicators	CRITIC	Entropy	Combined Weights
D11	0.0474	0.041	4.42%
D12	0.0298	0.0334	3.16%
D13	0.0290	0.0493	3.92%
D14	0.0291	0.0501	3.96%
D15	0.0280	0.0262	2.71%
D21	0.0326	0.0692	5.09%
D22	0.0261	0.0436	3.49%
D23	0.0967	0.1136	10.52%
D24	0.0869	0.0261	5.65%
D25	0.0299	0.0369	3.34%
D26	0.0257	0.0288	2.73%
D27	0.0499	0.0241	3.70%
D31	0.0284	0.0444	3.64%
D32	0.0305	0.0629	4.67%
D33	0.0271	0.0261	2.66%
D34	0.0372	0.0233	3.03%
D35	0.0294	0.0437	3.66%
D36	0.0401	0.0874	6.38%
D41	0.0333	0.0605	4.69%
D42	0.0243	0.0259	2.51%
D51	0.1299	0.0329	8.14%
D52	0.0531	0.0203	3.67%
D53	0.0555	0.0302	4.29%

.

Based on the combined weight values in the above table, the indicator dataset used for the comprehensive evaluation of TOPSIS was calculated according to Equation (10), and the positive and negative ideal distances D+ and D− and the relative proximity C of the green and digital economy were calculated according to Equations (11)–(13). The results obtained were as follows (see Figure 4 and Figure 5 for details).

4.2. Green Economy and Digital Economy Forecast

In this paper, we first used GM (1,1) to forecast data for the green economy and digital economy for the next five years; for the forecast results and forecast error table, see

Table 3. Grey prediction model prediction results.

Year	Digital Economy Predict	Forecast Data	Residuals Error	Simulation Error	Green Economy Predict	Forecast Data	Residuals Error	Simulation Error
2008	0	0	0	0	0.261	0.261	0.000	0.000
2009	0.11	0.178	−0.068	0.621	0.241	0.319	−0.078	0.324
2010	0.2	0.203	−0.003	0.016	0.269	0.343	−0.074	0.275
2011	0.293	0.231	0.062	0.210	0.336	0.369	−0.033	0.097
2012	0.321	0.264	0.057	0.179	0.403	0.396	0.007	0.017
2013	0.254	0.3	−0.046	0.183	0.447	0.426	0.021	0.048
2014	0.384	0.342	0.042	0.109	0.489	0.458	0.031	0.064
2015	0.335	0.39	−0.055	0.164	0.527	0.492	0.035	0.067
2016	0.429	0.444	−0.015	0.036	0.55	0.529	0.021	0.039
2017	0.529	0.506	0.023	0.043	0.591	0.568	0.023	0.039
2018	0.589	0.577	0.012	0.021	0.646	0.611	0.035	0.055
2019	0.655	0.657	−0.002	0.003	0.69	0.656	0.034	0.049
2020	0.738	0.749	−0.011	0.014	0.729	0.705	0.024	0.033
2021	0.822	0.853	−0.031	0.038	0.722	0.758	−0.036	0.050
2022	1	0.972	0.028	0.028	0.731	0.815	−0.084	0.114
2023E	-	1.107	-	-	-	0.875	-	-
2024E	-	1.262	-	-	-	0.941	-	-
2025E	-	1.437	-	-	-	1.011	-	-
2026E	-	1.638	-	-	-	1.087	-	-
2027E	-	1.866	-	-	-	1.168	-	-

and Figure 6 and Figure 7 for details. The average simulation relative errors for the green and digital economies were calculated to be 9.07% and 11.882%. The average simulation relative errors for the green economy and digital economy forecasts, calculated by summing the simulation error values from 2009 to 2022 and dividing by 14 years, were found to be 9.07% and 11.882%, respectively. To optimize our model, we derived the optimization coefficients ‘k’ by subtracting these average simulation relative errors from 100%. For the green economy, the optimization coefficient ‘k’ was 0.9093, while for the digital economy, it was 0.8812. So, the optimization coefficients k were 0.9093 and 0.8812, respectively.

Then, the LSTM model was used for training, where the neural network part of the model was set to two layers and the hidden layer was set to 40 neurons and trained 200 times with a learning rate of 0.01. When the model training results ESM met the demand, the same predicted the composite index for the next 5 years with the results which can be seen in detail in Figure 8 and Figure 9.

Finally, the predicted values of the GM model were optimized using the predicted values of the LSTM to obtain the final prediction results (see

Table 4. Predicted results.

Year	G-Grey Forecast	G-LSTM Forecast	G-Comprehensive Forecast Results	G-Grey Forecast	G-LSTM Forecast	G-Comprehensive Forecast Results
2023E	0.875	0.715	0.860	1.107	1.079	1.104
2024E	0.941	0.728	0.922	1.262	1.124	1.246
2025E	1.011	0.739	0.986	1.437	1.161	1.404
2026E	1.087	0.748	1.056	1.638	1.190	1.585
2027E	1.168	0.756	1.131	1.866	1.214	1.789

for details).

4.3. Green Economy and Digital Economy Synergy

According to the grey absolute, this paper establishes the correlation analysis of the green economy and the digital economy based on the absolute grey correlation model. The synergy degree of each year is found through the original and cumulative series of green economy and digital economy. The results can be seen in detail in Figure 10.

5. Discussion

5.1. Digital Economy and Green Economy Development Index

We used the TOPSIS method to evaluate the combined development levels of the digital and green economies. The analysis indicates continuous growth in the green economy, starting from 0.261 in 2008 to 0.731 in 2022. The most significant increase occurred between 2011 and 2012, rising from 0.336 to 0.403. The digital economy has also grown and accelerated since 2008, except for decreases in 2013 and 2015. It had the fastest growth rates of approximately 11.2% and 12.7% in 2017 and 2018, respectively. However, in 2019 and 2020, the growth rate of the digital economy index slowed down slightly due to the impact of the COVID-19 pandemic and the full release of the “consumer Internet” dividend. From 2020 to 2022, the growth rate of the digital economy index increased again, indicating that the digital economy’s advantages such as “contactless” and O2O effectively compensated for the shortcomings of the real economy during the pandemic. This further expanded the development space and became a crucial driving force for global economic growth.

5.2. Green Economy and Digital Economy Trend Forecast

The forecast predicts that the digital economy index will grow from 0.860 in 2023 to 1.131 in 2027, indicating significant growth potential. The “industrial Internet” will be a crucial direction for the digital economy’s future development. The forecast also suggests that the digital economy will continue to surpass the green economy and indirectly promote green economy growth by disseminating information and the best practices related to sustainable development. The continuous innovation and development of digital technology will further drive the growth of the green economy. However, it is important to strengthen digital security and privacy protection for the development of the digital economy. The growth rate of the digital economy will continue to outpace that of the green economy. Therefore, as the digital economy becomes more entrenched in society, it is crucial to prioritize green and sustainable development to foster deep integration and development between the digital and green economies.

5.3. Synergistic Evolution Law of the Green Economy and the Digital Economy

The analysis of historical data indicates that the synergy between the green economy and the digital economy increased from 2008 to 2018, reaching its peak in 2018 with a value of 0.995. However, from 2019 to 2022, the synergy index gradually decreased to 0.855 in 2022. By utilizing the grey prediction model and the LSTM model, the synergy between the green and digital economies is predicted to slightly decrease to 0.804 in 2023, followed by a recovery to 0.925 in 2024. From 2025 to 2027, the synergy is expected to gradually decline with values of 0.914, 0.900, and 0.885, respectively. This decline may be attributed to China’s implementation of strategies in the clean energy, energy conservation, emission reduction, and “double carbon” areas. The disorderly expansion of capital in the digital economy subsystem, particularly in the “platform economy,” has led to a period of structural adjustment. The shift in strategic priorities and the emergence of new policies have profoundly impacted the synergistic development of the two economic subsystems. The decrease in synergy from 2019 to 2022 primarily stemmed from the change in policy orientation favoring the development of the green economy subsystem over the digital economy subsystem, highlighting the differences in development connotations between the two economies. The migration of China’s economy from labor- and capital-intensive models to knowledge- and human-capital-intensive models consistently enhances the endogenous dynamics of the two economic subsystems, which in turn affects their synergy. The fluctuation of the synergy index may also reflect the impact of internal industrial restructuring and recombination of the two subsystems on their synergistic evolutionary effects. Additionally, the COVID-19 epidemic has had a significant impact on global trade, supply chains, and economic growth, leading to a reduction in the synergistic evolution process of the green and digital economies and affecting the level of synergistic evolution between the two economic subsystems.

5.4. Implications of Synergistic Evolution Trends

The observed trends in the synergistic evolution of the green and digital economies hold significant implications for policy, industry, and global economic dynamics.

5.4.1. Policy Considerations

The decrease in synergy from 2019 to 2022, driven by a shift in policy orientation favoring the green economy, underscores the profound impact of policy decisions on these economic subsystems. Governments worldwide are increasingly recognizing the importance of sustainable development and environmental conservation. Policies supporting clean energy, energy conservation, emission reduction, and the pursuit of “double carbon” goals have become central in the policy landscape. These shifts reflect the broader global commitment to addressing environmental challenges, including climate change. As the green economy takes precedence in policy agendas, it highlights the need for targeted policies that balance environmental priorities with economic growth and digital innovation. Policymakers must navigate the complex interplay between these two domains to ensure that they complement each other effectively.

5.4.2. Industry Dynamics and Structural Adjustment

The disorderly expansion of capital, particularly within the “platform economy” of the digital economy subsystem, has led to a period of structural adjustment. This adjustment phase is characterized by evolving business models, changing market dynamics, and a recalibration of industry priorities. Companies are reevaluating their strategies to align with new policies and shifting consumer preferences, which can impact their approach to sustainability. In this context, fostering a culture of responsible and sustainable business practices within the digital economy is vital. Companies should consider their environmental footprint, invest in green technologies, and explore synergistic opportunities with the green economy. This alignment can lead to innovative solutions that benefit both economic and environmental objectives.

5.4.3. Global Economic Resilience

The COVID-19 pandemic has underscored the interdependence of global economies and supply chains. It has affected the level of synergistic evolution between the green and digital economies. The pandemic-induced disruptions have necessitated a reevaluation of global economic resilience and risk management strategies. The projected recovery in synergy from 2023 to 2024 highlights the potential for adaptation and resilience in the face of external shocks. It suggests that as economies adjust to post-pandemic realities, they may find ways to enhance synergies between the green and digital sectors, creating more robust economic systems that can withstand future challenges.

In conclusion, the dynamic relationship between the green and digital economies reflects the evolving landscape of global economic development. The interplay between policy, industry dynamics, and global events like the COVID-19 pandemic shapes the synergistic evolution of these economic subsystems. Recognizing the interconnectedness of these domains and fostering a balanced approach to development will be essential for sustainable economic growth in the years ahead.

6. Limitations and Recommendations

To further promote the synergistic development of the green economy and the digital economy, the following suggestions can be considered:

• The organic connection with the “Digital China” strategy should be strengthened and the comprehensive integration of the green economy and the digital economy should be promoted. The development and adoption of digital technologies that optimize the production, distribution, and consumption of renewable energy should be encouraged, such as smart grid management systems, virtual power plants, and energy storage technologies. The investment in science and technology innovation in the field of digital technology and green environmental protection development should be increased, and breakthroughs in key technologies in the green digital field should be accelerated, such as energy-saving computing and AI-driven environmental monitoring. The integrated development of “government, industry, academia, research, and application” should be promoted to accelerate the promotion and application of green digital technologies. The ecosystem for cultivating innovation and entrepreneurship institutions should be improved in the green digital cross-sector and should guide professional investment institutions, third-party service institutions, results-incubation institutions, and other market entities to increase the support and services for startups in the green digital cross-sector field. We also suggest establishing and improving the system of laws, regulations, and standards in the green digital field, lowering the “threshold” and “barriers” for the synergistic development of the digital economy and green economy. Big data analysis and AI-driven insights should be actively utilized to provide an objective basis and strong support for policies and decisions related to environmental protection and sustainable development.
• The “double carbon” development goal should be the focus and the sustainable green development of key digital technologies and infrastructure should be enhanced. Supporting the construction of green data centers, the deployment of smart grids, and the development of energy-efficient broadband networks should be a priority. The proportion of green digital infrastructure should be increased, with digital government construction and digital economy headquarters enterprises as the focus. The characteristics and advantages of energy prices, climate, and geographical location in western China should be considered and energy-intensive data processing infrastructure should be deployed in areas with lower energy prices and better climate conditions to improve the efficiency of spatial allocation for green digital infrastructure development. The cultivation of new engines of economic and industrial development in less developed areas of western China should be promoted. Financial support modes should be introduced for the development of green digital technology and industry to expedite the industrialization and application of related technologies. For indirect financing, mainstream debt financing models should be used at home and abroad and tools such as special bonds, supply chain financing, PPP, asset-backed bonds, corporate bonds, and debt REITs should be adopted to provide financial support for major digital infrastructure projects. For direct financing, industry platform companies should be encouraged to diversify financing through debt-to-equity swaps, preferred shares, warrants, equity REITs, and other methods. Priorities should include strengthening talent cultivation in the field of green digital technology; encouraging research institutes, colleges and universities, and leading enterprises to increase the cultivation and recruitment of high-level talents in this field; improving the training mechanism for professional ability and management skills in green digital technology; and continuously enhancing talent levels.

7. Conclusions

This paper examines the synergy between the green and digital economies in China from 2008 to 2022 and predicts the synergistic evolution process from 2023 to 2027. The following conclusions can be drawn:

• The deep integration of the green economy and the digital economy is a trend for future development. The synergistic development of these two economic subsystems will strongly drive sustainable economic growth. This growth will be fueled by the continuous expansion of digital technologies like artificial intelligence, machine learning, and the Internet of Things, as well as increased reliance on digital platforms and services. The synergy between the green economy and the digital economy will become more prominent, with them playing vital roles in mutual promotion and support.
• The development level of the green economy and the digital economy will mutually benefit each other and continue to grow steadily. Forecast data suggest that both the digital economy and the green economy will experience growth in the next five years, with the digital economy expected to have a faster growth rate. The growth of the digital economy can contribute to sustainable development and facilitate the transition to a green economy. Meanwhile, the growth rate of the green economy appears relatively stable, with continuous development driven by increased global awareness of environmental issues, policy shifts towards sustainable development, and advancements in renewable energy and clean technologies.
• This paper discusses the suitability of LSTM (long short-term memory), a type of recurrent neural network, for forecasting time series data. It notes that LSTM performs well in matching real results but may not perform as well in forecasting preliminaries. On the other hand, GM (1,1), a grey model, fits the preliminary data better. Therefore, this paper suggests that combining these two methods can lead to more accurate time series predictions.
• This paper predicts that the synergy between the green economy and the digital economy will initially rise but may decline in the future. This decline is attributed to the potential competition for resources, such as investment, labor, and government support, which could hinder the realization of synergies. Regional disparities in growth rates between the two sectors and discrepancies in access to digital infrastructure and skills among different groups may also limit the integration and synergy. Additionally, mismanagement of digital technologies could lead to increased energy consumption and e-waste, negatively impacting the environment and undermining potential synergies.
• This paper concludes by emphasizing the substantial potential for synergies between the digital and green economies. It suggests that future research should focus on collecting sufficient sample data to accurately assess the evolving patterns of synergy between these two economic subsystems. Additionally, exploring the nonlinear relationships and interaction mechanisms between the green and digital economies is deemed crucial. The impact of related measures and policies on the development of these economies should also be considered, and certain factors should be identified to label training samples for LSTM models to enhance research accuracy.

The synergy between the green economy and the digital economy is projected to rise initially and then decline in the future. The expansion of these two sectors may lead to competition for resources such as investment, labor, and government support, which can hinder the full realization of synergies. Additionally, different regions or sectors of China may experience varying growth rates in the green and digital economies, resulting in imbalanced development and reduced integration, thereby limiting potential synergies. Moreover, disparities in access to digital infrastructure, technologies, and skills among different social groups or regions can impede the achievement of synergies between the two economic subsystems. Furthermore, the mismanagement of digital technologies can contribute to increased energy consumption and e-waste, negatively impact the environment, and undermine potential synergies.

In general, there is substantial potential for synergies between the digital economy and the green economy. Future research should ensure that the sample data are sufficient to accurately assess the synergistic evolution patterns between the two economic subsystems. Additionally, exploring the nonlinear relationship and interaction mechanisms between the green economy and the digital economy is crucial. The development of the digital economy and the green economy is likely to be influenced by related measures and policies, and future research should consider screening certain factors to label training samples for LSTM models to enhance the research.