Research and analysis on the coordination mechanism of financial innovation and economic growth based on BP neural network

Abstract

There is little research on the relationship between financial innovation and economic growth, and the research on the synergy between the two is basically blank. Based on this, from a general perspective, through constructing the corresponding subsystems in combination with financial innovation and economic growth, establishing the corresponding synergy model, and discovering the synergy development relationship by studying the degree of synergy in the past period, this study builds a BP neural network simulation model to predict the degree of synergy between financial innovation and economic growth in 2018 on the basis of practice. At the same time, this study compares it with the actual situation to verify its effectiveness. Through analysis, the research model has certain effectiveness, which is basically consistent with the actual development trend. The research proposes that the main trend of financial innovation from the perspective of generalized virtual economy is Internet finance. This is the first time to study this issue from a new perspective, theory and method, which expands the existing research results.

Keywords

BP neural network financial innovation economic growth synergy

1 Introduction

The relationship between financial innovation and economic growth in the new economic form must be complicated. A series of questions worth pondering is: From the perspective of the generalized virtual economy, what is the relationship between financial innovation and economic growth, and how the two will work. The degree of synergy between them will affect the formulation of economic and financial policies and the development of the global economy. Combining the characteristics and theories of the generalized virtual economy era and studying the degree of synergy between financial innovation and economic growth from the perspective of generalized virtual economy has important theoretical and practical significance.

In the context of the rapid development of the generalized virtual economy, we are faced with an increasingly tight and complex reality between financial innovation and economic growth. This paper explores the research on the synergy mechanism between financial innova-tion and economic growth from the perspective of generalized virtual economy and strives to make breakthroughs in theory and enrich the theory of the relationship between financial innovation and economic growth. The global economy has now entered a broad-based virtual economy era in which all aspects of economic activity permeate into almost all areas of economic activity, with the aim of satisfying people’s psychological needs and spiritual needs, and reflecting the elements of brand, service, experience and cultural consumption. This requires exploring new economic theories, new development paths, new growth models, and new policy developments. The study of the relationship between financial innovation and economic growth from the perspective of generalized virtual economy is aimed at enriching the research results in the field of generalized virtual economy in the new economy and improving the economic and financial system. Strengthening the study of generalized virtual economic theory can solve the transformation of economic growth model, formulate new economic and financial policies, and provide theoretical basis for studying new ideas and new models of the relationship between financial innovation and economic growth. The virtual economic phenomenon and the earth-shaking changes more than a hundred years ago. For example, value virtualization and capital securitization are an extremely important feature of today’s social and economic activities. These new economic phenomena have not been described by traditional economic theories, which leads to the traditional economic growth theory not able to explain the causes of economic growth in some developed countries and even developing countries at this stage. Therefore, the research in this paper will verify the generalized virtual economic theory and enrich the results of generalized virtual economy, which has strong theoretical value.

In recent years, with the rapid development of financial innovation, more and more finan-cial instruments such as options, futures and other financial derivatives have been widely rec-ognized and used. At the same time, financial innovation has improved the overall efficiency of the financial system, which has caused fundamental changes in the traditional financial industry. The innovation of the securities market promoted the development of bonds and stock markets, the rise of financial engineering led to the development of financial derivatives markets, and the technological revolution in the financial industry made the virtual economy grow rapidly. However, these are all in the narrow sense of the virtual economy. It can be said that the emergence of a generalized virtual economy makes the economic field more comprehensive and perfect and makes the relationship between financial innovation and economic growth in the new economy more complicated and diversified. Governments should note that in the new economic era, it is imperative to study the relationship between financial innovation and economic growth as soon as possible, and to recognize the importance and great potential of the relationship between the two. Only by standing on the forefront of the times, advancing with the times, absorbing new products of the times, and constantly enriching, developing and perfecting existing economic theories, can we explore the policy recommendations for a virtuous cycle of financial innovation and economic growth in the new economic environment, so that it can better serve the development of the entire human society. In time, when China’s generalized virtual economy develops and develops, it will certainly have a far-reaching practical significance to study the relationship between financial innovation and economic growth from the perspective of generalized virtual economy.

2 Related work

There are very few literatures on the topical study of the relationship between financial innovation and economic growth at home and abroad, but there are many research results related to this. These research results mainly focus on two aspects, one is to explore the relationship between financial innovation and financial development, and the other is to explore the relationship between financial development and economic growth. When discussing the relationship between financial innovation and financial development, most scholars have acknowledged that financial innovation has effectively promoted financial development. For example, Boot&Thakor believes that financial innovation is an important force in the evolution of the financial system [1]. Ireland’s research has changed the past view of the instability of the money demand function to financial innovation in the private financial sector. By establishing a theoretical model of money demand that joins financial innovation, empirical evidence shows that financial innovation is conducive to the stability of money demand [2]. Zhao Hemin believes that financial innovation is the soul of sustainable financial development, a new type of financial instrument that meets the needs of financial development and economic development. The continuous emergence of financial products plays an important role in the rational and effective development and utilization of financial resources to ensure the sustainable development of the financial industry [3]. While Chen Ziji pointed out that financial innovation is a “double-edged sword”, it also fully affirmed the role of financial innovation in promoting financial development [4].

The study of the relationship between financial development and economic growth began in the 1960s. In terms of theoretical research, Goldsmith summarizes various financial phe-nomena into three basic aspects, namely financial instruments, financial institutions, and financial structures. He believes that the more developed the finance, the stronger the penetration of financial activities into the economy, and the faster the economic growth will [5]. Mckinnon gave up the hypothesis that money and capital are replaced by each other in traditional financial theory. He believes that the backward financial system of developing countries makes investment not rely on external financing but relies on internal financing [6]. Shaw put forward the “debt media theory”, which believes that money is a medium in the financial system, not a real social wealth. Money plays a variety of media roles throughout society, increasing production efficiency, increasing output, and promoting savings and investment by reducing production and transaction costs [7]. King and Levine began to study the impact of financial development on economic growth from the perspective of financial function [8]. Rajah and Zingales carefully analyzed the micro-effect mechanism of finance for economic growth [9]. After this, financial scientists began to use different measurement methods and different sample data to test the correlation between financial development and economic growth. For example, Neusser and Kugler and Rousseau and Wachtel use time series data, Beck, Levine and Loaya use panel data, and Levine uses instrumental variables to study the relationship between financial development and economic growth [10]. Adolfo Sachsida’s time series data and Granger causality test results show that financial development and economic growth are mutually causal [11]. The panel data used by Beck and Levine, NomanLoayza and RomainRanciere, and the results of the GMM (generalized moment estimation) approach indicate that financial deepening has a significant positive effect on economic growth. YousifKhalifaAI.Yousif, Ce7satCaldero’n&LinLiu, etc. use empirical data from developing countries to demonstrate that there is a strong two-way causal relationship between financial development and economic growth [12].

There are also many scholars in China who have studied the relationship between eco-nomic growth and financial development. Shi Yongdong, Wu Zhi, etc. using Granger causality test and econometric analysis based on the Cobb-Douglas production function framework show that there is a two-way causal relationship between China’s financial development and economic growth [12]. Han Tingchun established a model of the endogenous mechanism of interaction between financial development and economic growth, and based on mathematical analysis, obtained equilibrium conditions for sustained economic growth. He believes that as long as the efficiency of the financial sector continues to increase, and the level of intangible capital continues to grow, the sustainable development of the economy becomes possible [13]. Zhao Zhenquan and Xue Fenghui analyzed the effect of China’s financial development on economic growth by using the revised output growth rate model of Greenwoal-Jovanovic model. The results show that between 1994 and 2002, China’s credit market has a significant effect on economic growth, and the role of the stock market is not obvious [14].

In short, there are only a handful of collaborative research in the economic field, and there is very little research on the relationship between financial innovation and economic growth. Therefore, using this theory to study the relationship between the two is innovative.

3 Theoretical analysis

3.1 BP neural network

BP (BackPropagation) neural network is the most mature neural network method applied in academic circles so far, and its application is mainly distributed in the fields of intelligent control and information processing. The algorithm is based on two directions, one is a multi-level neural network that is forward-conducting, and the other is a back-propagation that encounters errors. The BP network consists of an input layer, a hidden layer and an output layer, wherein the hidden layer is one or more layers, and its network structure is as shown in Fig. 1 [15]. The signal is generally forward conduction. If the actual output result is different from the expected output result in the process, the error propagates in the opposite direction, and the information is equally distributed to each unit of each layer to obtain the error information of all the units. By adjusting the weights between the nodes of each layer, the output error is controlled within the required standard [16].

Fig.1

BP neural network structure image.

The BP neural network algorithm is mainly divided into: forward propagation of signals and back propagation of errors. The specific performance is that the signal experiences the forward propagation of the input layer, the hidden layer, and the output layer. At the same time, the existence of the error causes the signal to be selected for back propagation, that is, the direction of signal propagation is the output layer - the hidden layer - the input layer. Analysis of the BP neural network structure diagram in Fig. 1 can obtain the quantitative relationship between each layer. The meaning of the relevant variables is shown in Table 1 [17].

Table 1

Variable meaning of BP neural network

Symbol	Description	Symbol	Description
X _i	represents the input of the i-th node of the input layer, where i = 1,...,m;	ψ _(x)	represents the excitation function of the output layer
W _ij	represents the threshold of the j-th node of the input layer	o _k	represents the output of the k-th node of the output layer
θ _ij	represents the excitation function of the hidden layer	y _i	the output of the first node of the hidden layer
φ (x)	represents the excitation function of the hidden layer	net _k	Input of the kth node of the output layer
W _jk	Weight between the j-th node of the hidden layer and the k-th node of the output layer£¬j=1,..,q	E	Reverse error
a _k	represents the threshold of the k-th node of the output layer£¬K=1, ... ,L	d _k	Actual output of the k-th node of the output layer

(1) Forward propagation of signals

The signal propagates from the input layer to the output layer through the hidden layer, and the expression of the output of the kth node of the output layer is obtained by the mathematical relationship existing between the layers: $\begin{matrix} o_{k} = ψ ({net}_{k}) = ψ (\sum_{j = 1}^{q} w_{jk} y_{j} + a_{k}) = \\ ψ (\sum_{j = 1}^{q} w_{jk} φ (\sum_{i = 1}^{m} w_{ij} x_{i} + θ_{j}) + a_{k}) \end{matrix}$ (1)

(2) Back propagation of error

In the neural network structure, when the output signal of the output layer does not match the expected signal value, an output error E is generated. At the same time, the direction of propagation of the signal is reverse propagation. From the output layer through the hidden layer to the input layer, the output error of each level of neurons is calculated one by one, and the error steepest descent method is selected to correct the weight of each level. Eventually, the actual output value of the neural network structure after adjustment is as close as possible to the expected value. If the number of training samples is P, then the overall error criterion function of the system for P training samples is: $E = \frac{1}{2} \sum_{p}^{p} \sum_{k = 1}^{1} {(T_{k}^{p} - o_{k}^{p})}^{2}$ (2)

According to the error steepest descent method, the correction amount of the output layer weight Δw_jk is adjusted one by one, and the correction amount of the hidden layer weight Δw_ij is adjusted one by one. Its expression is shown in formula (3). Among them, η represents the learning efficiency, which is a constant term, and the interval is between 0 and 1. ${\begin{matrix} Δ W_{jk} = - η \frac{\partial E}{\partial w_{jk}} \\ Δ W_{ij} = - η \frac{\partial E}{\partial w_{ij}} \end{matrix}$ (3)

Then, E is substituted into formula (3), and the weights and threshold correction formulas of the following three-layer BP feedforward network learning algorithm can be obtained. ${\begin{matrix} Δ W_{jk} = η \sum_{p = 1}^{p} \sum_{k = 1}^{1} (T_{k}^{p} - o_{k}^{p}) ψ ({net}_{k}) y_{i} \\ Δ W_{ij} = η \sum_{p = 1}^{p} \sum_{k = 1}^{1} (T_{k}^{p} - o_{k}^{p}) ψ ({net}_{j}) w_{jk} φ ({net}_{j}) x_{i} \end{matrix}$ (4)

By using the BP neural network model, this paper constructs a time-based sequential parameter lateral prediction model and a longitudinal synergy prediction model based on time and order parameters to predict the order parameters and the degree of synergy of financial innovation and economic growth system under the generalized virtual economy.

3.2 The degree of order model of financial innovation and economic growth subsystem

We set up a complex system S of financial innovation and economic growth from the perspective of generalized virtual economy. The components of S include financial innovation subsystem and economic growth subsystem S_j, j [1, 2]. Subsystem S_j = (S_j1, S_j1, . . . . . . , S_jm). The order parameter synergy of the subsystem S_j determines state and structure of S. We set the order parameter of financial innovation and economic growth in the development process can be expressed as e_j = (e_j1, e_j2, …, e_jm) , m ≥ 1, β_ji ≤ e_ji ≤ α_ji, i ∈ [1, m]. One of the situations is that the larger the value of each parameter in e_jl+1, e_j2, . . . . . . e_jl (j = 1, 2), the higher the degree of ordering of the subsystem. Conversely, if the value of each parameter is smaller, the degree of order of the subsystem is lower. Another situation is that the larger the value of each parameter in e_j1+1, e_j2, . . . . . . e_jm (j = 1, 2) is, the lower the degree of order of the system is. In other words, the smaller the value of each parameter is, the higher the degree of order of the system is. The degree of order of the order parameter component e_ji of the financial innovation subsystem and the subsystem S_j (j = 1, 2) of economic growth can be expressed as [18]: ${u_{j} (e_{ji}) = {\begin{matrix} \frac{(e_{ji} - β_{ji})}{(α_{ji} - β_{ji})}, i \in [1, l] \\ \frac{(e_{ji} - β_{ji})}{(α_{ji} - β_{ji})}, i \in [l + 1, m] \end{matrix}$ (5)

Among them, α_ji and β_ji represent the maximum and minimum values of the j-th system on the i-th indicator, respectively. As can be seen from the above formula, u_j (e_ji) ∈ [0, 1].When the value of u_j (e_ji) is larger, the greater the orderly contribution of e_ji to the system. In general, the “total contribution” of the order parameter variable e_ji to the degree of order of the sub-mechanism S_j can be realized through integration of u_j (e_ji). The form of integration depends on the value of each order parameter and the combination between subsystems. There are usually two methods of summation, which are collective mean or linear weighting. That is, it can be expressed as $u_{j} (e_{j}) = \sqrt[m]{{coprod}_{i = 1}^{m} u_{j} (e_{ji})}, j = [1, 2]$ (6)

Secondly, it can also be expressed as: $u_{j} (e_{j}) = \sum_{i = 1}^{m} ω_{i} u_{i} (e_{ji}), ω_{i} \geq 0, \sum_{i = 1}^{m} ω_{i} = 1$ (7)

This paper uses the linear weighting method to determine the order function of the financial innovation and economic growth subsystem. In the method of linear weighted summation, in addition to considering the actual state of the whole system, the weight of the composite system needs to reflect the expected realization goal of the composite system over a period of time. In other words, weight is the role or possession of e_ji ∈ (0, 1) in maintaining the orderly progress of the system. It can be known from formula (7) that the larger the value of e_ji ∈ (0, 1), the higher the degree of order of the subsystem; on the contrary, the lower the degree of order of the subsystem [19].

3.3 Collaborative Measurement Model of Financial Innovation and Economic Growth Subsystem

When the initial time t₀ is determined, the degree of order of the financial innovation subsystem and the economic growth subsystem order parameter is $u_{j}^{0} (e_{j})$ . At the moment $u_{j}^{1} (e_{j})$ in the development process of the complex system, the degree of order of the financial innovation subsystem and the economic growth subsystem order parameter is $u_{j}^{1} (e_{j})$ . The coordination degree between the definition of financial innovation subsystem and economic growth subsystem is defined as [20]: $syn = η \sum_{j = 1}^{m} δ_{j} | u_{j}^{1} (e_{j}) - u_{j}^{0} (e_{j}) |$ (8)

Among them, $η = \frac{min [u_{j}^{2} (e_{j}) - u_{j}^{0} (e_{j}) \neq 0]}{| min [u_{j}^{2} (e_{j}) - u_{j}^{0} (e_{j}) \neq 0] |}$ £¬j = 1, 2, . . . , m£¬δ_j ≥ 0£¬ $\sum_{j = 1}^{m} δ_{j} = 1$ . It can be known from formula (3.3) that the higher the degree of synergy between the financial innovation and the economic growth complex system, the better the degree of synergy of the whole system. Conversely, the lower the degree of synergy value of the whole system, the worse the degree of coordination. Apparently, syn ∈ [0, 1]. S_j is the degree of change in the systematic order of subsystem S_j from the initial moment to a selected time. It characterizes how much the subsystem becomes more orderly from the initial moment to a selected moment. The meaning of η is that the system has positive synergy only when all $u_{j}^{1} (e_{j}) - u_{j}^{0} (e_{j}) > 0$ are established, anyone $u_{j}^{1} (e_{j}) - u_{j}^{0} (e_{j}) \leq 0$ . The degree of synergy..is manifested by the degree of reverse development of the two subsystems or not at all [21].

The degree of coordination of the entire composite system depends on the order of the financial innovation subsystem and the economic growth subsystem. That is to say, if the order value of the financial innovation subsystem increases more, the degree of order of the economic growth subsystem increases less, or even increases or decreases. Then the coordination of the entire composite system is poor or there is no coordination, and vice versa. Equation (8) also illustrates the change in the order of the order parameters of the two subsystems during the study period. Based on this degree of change, the dynamic synergistic process of the composite system can be inferred [22].

3.4 Basic description of the collaborative prediction model of financial innovation and economic growth

The degree of synergy between financial innovation and economic growth is an important indicator for measuring the healthy development of the financial sector in the broad virtual economy. The synergy between financial innovation and economic growth depends on the value of each order parameter of the system. Therefore, the prediction of system synergy needs to be based on the prediction of the system’s order parameters. In the process of analyzing financial innovation and economic growth system, it is found that the synergy system between the financial innovation and economic growth is a dynamic superposition system. That is to say, the degree of order and the degree of synergy of the system depend not only on the values of the order parameters of the system in the current year, but also on the values of the order parameters of the system in previous years. Using the previous data and calculation methods, the systematic order and degree of synergy between financial innovation and economic growth in 2012-2017 and 2013-2018 are analyzed separately, as shown in Figs. 2, 3 and Table 2.

Fig.2

Degree of order and degree of synergy curve of financial innovation and economic growth during 2012-2017.

Fig.3

Degree of order and degree of synergy between financial innovation and economic growth during 2013-2018.

Table 2

Degree of order and system synergy of subsystems during 2012-2017, 2013-2018

System interval	Year	Financial innovation order	Economic growth order	Degree of synergy
2012-2017	2012	0.278	0.08	— —
	2013	0.399	0.234	0.148
	2014	0.402	0.452	0.258
	2015	0.357	0.491	0.255
	2016	0.941	0.628	0.616
	2017	0.847	0.89	0.7
2013-2018	2013	0.242	0.18	— —
	2014	0.448	0.377	0.212
	2015	0.259	0.297	0.078
	2016	0.882	0.387	0.434
	2017	0.733	0.612	0.472
	2018	0.636	0.774	0.504

According to the analysis of Figs. 2, 3 and Table 2, it can be seen that in different system intervals, the degree of financial innovation order, economic growth order and degree of synergy in the same year are different. Based on the above considerations, this paper proposes to establish a time-based horizontal parameter prediction model and a longitudinal cooperative prediction model based on time and order parameters synthesis. The basic structure of the model is shown in Fig. 4. According to the description of the model, the two models are described as follows [23]:

Fig.4

Basic structure image of the prediction model.

(1) Model one: Time-based sequential parameter lateral prediction model. The model is a predictive model based on the various parameters of financial innovation and economic growth from the perspective of generalized virtual economy. It uses the values of the order parameters in different years as the prediction sequence to predict the order parameters of the next year.

(2) Model two: Longitudinal collaborative prediction model based on time and order parameter synthesis. As can be seen from the above description, in calculating the system cooperation degree, although the selected order parameters are fixed, and the values of the order parameters are fixed, the different time intervals of the different systems are also different due to the different time intervals selected. That is, the system is a system that changes dynamically over time. Therefore, in the prediction, not only the order parameters of the system but also the dynamic variability of time should be considered.

4 BP neural network prediction model construction and solution

In the following, a specific BP prediction model is established based on the specific analysis object, namely the financial innovation and the order parameter of the economic growth system, and the detailed solution steps are given. In order to facilitate the formal description of the problem, the system parameters and indicators are described in Table 3 [24].

Table 3
Formal description of the order parameter

Variable name Description Variable name Description

Total financial assets/GDP TAssetsGDP Per capita disposable income of city residents* City’Income

M2/GDP* M2GDP Per capita net income of rural households* Rurallncome

Financial institution loan balance/GDP* BalanceGDP Financial innovation order parameter set Finlnv=[TAssetsGDPM2GDP

BalanceGDPTAssetsMl

InvestmentRDRDEmployee]

Total amount of financing/M1* TAssetsMl Economic growth order parameter set EcoGrw= [GrowthRateGDP Fiscaȼòkvenue Fixed Assets ConsumerGoodsGD Per Citylncome Rurallneome]

Financial industry R&D investment* InvestmentRD Financial innovation order parameter set Standardized FinXllv

Financial industry R&D personnel account RDEmployee Economic growth standardization parameter set Standardized EcoGrw

for the total number of employees in the financial industry*

GDP growth rate* GowthRateGDP Financial innovation subsystem weight set FinlnvWeight= [wlw2w3w4w5w6]

Revenue* FiscaIlkvenue Economic growth subsystem weight set EcoGrwWeight= ivlv2v3v4v5v6v7]

Fixed assets investment in the whole society* FixedAssets Degree of order of financial innovation subsystem FinInvOrder

The total retail sales of social consumer goods* ConsumerGoods Degree of order of Economic growth subsystem EcoGrwOrder

Per capita GDP* GDPef System synergy CoUaborative-Degree

Variable name	Description	Variable name	Description
Total financial assets/GDP	TAssetsGDP	Per capita disposable income of city residents*	City’Income
M2/GDP*	M2GDP	Per capita net income of rural households*	Rurallncome
Financial institution loan balance/GDP*	BalanceGDP	Financial innovation order parameter set	Finlnv=[TAssetsGDPM2GDP
BalanceGDPTAssetsMl
InvestmentRDRDEmployee]
Total amount of financing/M1*	TAssetsMl	Economic growth order parameter set	EcoGrw= [GrowthRateGDP Fiscaȼòkvenue Fixed Assets ConsumerGoodsGD Per Citylncome Rurallneome]
Financial industry R&D investment*	InvestmentRD	Financial innovation order parameter set	Standardized FinXllv
Financial industry R&D personnel account	RDEmployee	Economic growth standardization parameter set	Standardized EcoGrw
for the total number of employees in the financial industry*
GDP growth rate*	GowthRateGDP	Financial innovation subsystem weight set	FinlnvWeight= [wlw2w3w4w5w6]
Revenue*	FiscaIlkvenue	Economic growth subsystem weight set	EcoGrwWeight= ivlv2v3v4v5v6v7]
Fixed assets investment in the whole society*	FixedAssets	Degree of order of financial innovation subsystem	FinInvOrder
The total retail sales of social consumer goods*	ConsumerGoods	Degree of order of Economic growth subsystem	EcoGrwOrder
Per capita GDP*	GDPef	System synergy	CoUaborative-Degree

Note: The variables indicated by the* in the table are column vectors composed mainly of years

The basic solution steps of the BP neural network prediction model are shown in Figs. 4 –6. The specific steps are as follows:

Step 1: Data sorting. According to the China Statistical Yearbook, China Financial Yearbook, China Science and Technology Statistical Yearbook, and the Y1.Y2 year data released by the People’s Bank of China website, the financial innovation order parameter matrix Finlnv(n) and economic growth order parameters matrix EcoGrw(n) are compiled and input as system data. Among them, Y2.Yl+l—n, that is, it is necessary to sort out the data for 11 consecutive years.

Step 2: Data standardization. Finlnv(n) and EcoGrw(n) were normalized according to the method of Chapter 3 to form a standardized matrix StandardizedFinlnv(n) and StandardizedEcoGrw(n), respectively.

Step 3: Predicting the order parameters. StandardizedFinInv(n) and StandardizedEcoGrw(n) are input to the order parameter lateral prediction BP network Nets, thereby the result set FinInv(n + 1), EcoGrw(n + 1), Standard-izedFinlnv(n + 1), and StandardizedEcoGrw(n + 1) including the prediction result of the y2 + 1th order parameter are output

Step 4: Form a system ordering result set. Finlnv(n + 1), EcoGrw(n + 1), StandardizedFinlnv(n + 1), and StandardizedEcoGrw(n + 1) are imported into the Order Model of Financial Innovation and Economic Growth Order (see Chapter 3 for details). Thus, the order degrees FinlnvOrder(n + 1) and EcoGrwOrder(n + 1) of the two subsystems are respectively formed. It should be noted here that in order to ensure a sufficient amount of sample size, the input data sets FinInv(n + 1), FinGrw (n + job StandardizedFinlnv(n + 1) and StandardizedEcoGrw(n + 1) need to be serialized to form yl-yl+1,Yl-Yl+2, y1.yl+3,..., yl-y2 sequences in turn, which is used as input to the OrderModel.

Step 5: Order and synergy prediction. The result of step 4 is input into the longitudinal cooperative prediction BP network NetL of time and sequence parameters to obtain Y1.Y2 + 1 year system cooperation degree [25].

Fig.5

Forecast process image.

Fig.6

Comparison image between actual values and predicted values of financial assets/GDP during 2003-2017.

4.1 Empirical Analysis of Financial Innovation and Economic Growth System Forecast

This paper uses Matlab simulation software to construct BP neural network with financial innovation and economic growth order parameters. The basic prediction process is shown in Fig. 5. The 2012-2017 financial innovation and economic growth order parameters are selected as learning samples and training samples. The neural network training is first carried out, and then the prediction effect of the neural network is judged. The 2018 order parameter index is predicted in the paper.

Before starting the prediction work, the neural network model needs to pass the sample training, and the relevant training parameters are set at this time. It is determined that the number of iterations is 1000, the control error is 0.001, and the power factor a = 0.9, which can avoid the minimum value of the BP network part, and it is easier to correct the weight. In addition, the number of output layer nodes associated with the neural network is set to 1, the number of hidden layer nodes is 4, the activation function of the output layer is purelin, and the activation function of the hidden layer is logsig.

4.2 Financial innovation and economic growth order parameter prediction

The financial innovation subsystem has six order parameter indicators, and the economic growth subsystem has seven order parameter indicators. In order to simplify the problem, this paper does not consider the correlation between each order parameter and predicts each order parameter separately. In this paper, the corresponding order parameters are input into the BP network one by one for prediction, and the data of the previous three years is used to predict the data for the next year, thus completing the training of the neural network.

To illustrate the effectiveness of neural network prediction, this paper lists the actual output and predicted output curves to illustrate. Since there are 13 prediction order parameters and each prediction network is trained separately, the relevant chart is listed here only by taking the total financial assets/GDP as an example. The comparison image between the actual value and the predicted value of the neural network is shown in Fig. 6 and the specific data is shown in Table 4. The error training curve is shown in Fig. 7. As can be seen from Figs. 6, 7 and Table 4, the prediction results are basically consistent with the actual, which indicates that the neural network is effective, and the neural network output reaches a given error requirement when training 35 steps.

Fig.7

Error training curve of total financial assets/GDP during 2013-2017.

Table 4

Comparison values of actual and predicted values of neural network

	Actual value	Predictive value
2013	3.137	3.233
2014	4.147	4.047
2015	4.165	4.352
2016	4.024	4.124
2017	4.42	4.401

Table 5

Predicted values of various order parameters of the financial innovation and economic growth system in 2013

Financial Innovation	Total financial assets/GDP	4.606
	M2/GDP	1.948
	Financial institution loan balance/GDP	1.300
	Total financial assets/Ml	7.369
	Financial in industry R&D investment	444.059
	Financial industry R&D investment personnel account for the total number of employees in the financial industry	0.533
Economic Growth	GDP growth rate	8.086
	Revenue	154404.220
	Fixed assets investment in the whole society	524801.010
	social consumer goods	299085.850
	Per capita GDP	36543.798
	Per capita disposable income of urban residents	23687.859
	Per capita net income of rural households	6955.671

4.3 Degree of order and synergy prediction of financial innovation and economic growth

According to the output order parameters, the degree of order and of coordination of the financial innovation and economic growth subsystems from 2005 to 2013 can be determined. The predicted results are shown in Fig. 8, and the specific data is shown in Table 6.

Fig.8

Degree of order and system synergy curve of financial innovation and economic growth during 2010-2018.

Table 6

Degree of order and system synergy value of financial innovation and economic growth during 2010-2018

Year	Financial innovation order	Economic growth order	Degree of synergy
2010	0.354	0.185	–
2011	0.433	0.273	0.184
2012	0.461	0.394	0.258
2013	0.441	0.373	0.238
2014	0.948	0.433	0.521
2015	0.816	0.572	0.525
2016	0.768	0.687	0.558
2017	0.997	0.773	0.715
2018	0.937	0.906	0.752

As can be seen from Fig. 8, the 2010-2018 financial innovation and economic growth subsystem ordering curve is similar to the order of the two subsystems during 2010-2017, and the overall trend is increasing. The degree of synergy between financial innovation and economic growth system in 2010-2018 is similar to the trend of financial innovation and economic growth system synergy between 2010 and 2017, that is, both show growth trends and the difference is small. In order to maintain the degree of consistency between financial innovation and economic growth from the perspective of the generalized virtual economy, it is necessary to propose a coordinated development model.

5 Collaborative models of financial innovation and economic growth

The collaborative development model of financial innovation and economic growth is a complex interaction process. Internet finance and traditional financial innovation drive economic growth by reducing transaction costs, promoting technological advancement, improving innovation efficiency, enabling resource allocation, and meeting individualized user needs. At the same time, information on brands, culture, creativity, humanities and services has been integrated into economic growth. The economic growth that gives the generalized virtual economy connotation promotes financial innovation through four factors: financial innovation property rights incentives, financial innovators’ fair returns, financial innovation government forces, and financial innovation institutions internalincentives.

According to the view of complex science, because of the close connection and interaction between financial innovation and economic growth, the two are regarded as a complex system, which is a combination of financial innovation and economic growth subsystem. The financial innovation and economic growth subsystems together constitute the whole of interaction and influence. The coordinated development of financial innovation and economic growth from the perspective of generalized virtual economy will produce the overall synergy effect of “1 + 1 > 2”, and ultimately promote economic development and progress under the general virtual economy. The specific situation is shown in Fig. 9.

Fig.9

Synergy model of financial innovation and economic growth.

6 Conclusion

Based on the perspective of generalized virtual economy and the research of the relationship between financial innovation and economic growth, this paper, combined with the theory of generalized virtual economy and the theory of synergy, analyzes the development trend of financial innovation and analyzes the main trends of financial innovation in the era of generalized virtual economy. At the same time, according to the principles of rationality, operability, comprehensiveness and comparability, this paper selects the index system of financial innovation and economic growth subsystem and constructs a synergy mechanism model with two subsystem order parameters. In addition, this paper empirically analyzes the orderliness of the financial innovation and economic growth subsystem and the synergy of the composite system during 2010-2017. Finally, based on BP neural network model to predict the synergy between financial innovation and economic growth system in 2018, this paper proposes a synergistic model of financial innovation and economic growth from the perspective of generalized virtual economy. This paper sorts out the historical situation of each stage of financial innovation from a vertical perspective. At the same time, combined with the characteristics of the times and research theories of generalized virtual economy, this paper proposes that the main trend of financial innovation from the perspective of generalized virtual economy is Internet finance. This is the first time to study this issue from a new perspective, theory and method, and to expand the existing research results.

Footnotes

Acknowledgments

This work was supported by National Natural Science Foundation of China (No.51709116), the Key Scientific Research Projects of Henan Province Universities and Colleges (No.17B570003) and Foundation for Dr in North China University of Water Resources and Electric Power (No.10030). Special thanks for the reviewers and their constructive comments and suggestions in improving the quality of this manuscript.

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