Abstract
Deep learning is a breakthrough in machine learning research. It aims to establish a deep network structure that can simulate the human brain for analysis and learning, interpret data through the mechanism of layer-by-layer abstract feature representation, and has excellent feature learning capabilities. According to the input-output performance evaluation data of colleges and universities, three experiments are done. First, the feature expression ability of RBM, the basic building block of deep learning, is studied, and compared with PCA, the results show that RBM-fine-tuning has better performance than PCA-expressed classifier; the reconstruction error can be used to judge the hidden layer. As the number of RBM layers increases, the classification accuracy gradually increases, indicating the feasibility of the RBMs feature extractor. Second, the model in this study has a higher prediction accuracy than other classification models and clarifies the effectiveness of the modular deep learning model based on RBMs from the perspectives of network convergence analysis and network output analysis. The ability is stronger than DBN, and the obtained abstract feature representation is more conducive to classification. Although the classification accuracy rate of the model in this study has been improved, the model has certain limitations. The network initialization is still set based on experiments and experience, and the prediction accuracy rate is only 88.3%, which needs to be improved. The parameter training algorithm of RBMs can be further studied. To improve the more accurate reference basis for the performance evaluation of colleges and universities. Third, in the research of dynamical systems, the stability of the time-delay unified system at zero equilibrium and positive equilibrium is studied, and the conditions for generating the Hopf branch are given. At the same time, some conclusions are obtained through theoretical analysis. Numerical simulations further verify the validity of the theoretical results.
1. Introduction
Since the twentieth century, a large number of time-delay dynamics problems have appeared in many disciplines of natural science and social science, such as optical soliton communication, neural network, ecosystem, transportation schedules, and so on [1]. The evolution trend of the system depends not only on the current state of the system but also on the state of the system at a certain time or several moments in the past. We call this type of dynamical system a time-delay dynamical system. Strictly speaking, the time delay is usually unavoidable in dynamic systems, even in information systems that travel at the speed of light. In addition, unlike the dynamical system described by ordinary differential equations, the solution space of the time-delay dynamical system is infinite, and its theoretical analysis is often difficult. Therefore, the study of delay differential equations is a very challenging direction [2].
Time delay often causes the motion of the system to become unstable, resulting in various forms of bifurcations [3]. Among these bifurcations, the most discussed is bifurcation, and this is mainly caused by the change of parameters. The time-delay system also shows more complex dynamics behavior. Therefore, studying bifurcation is an important aspect of studying dynamics. Mathematical models of time-delay systems belong to the category of functional differential equations (4). The essential difference from ordinary differential equations is that functional differential equations describing time-delay systems are infinite-dimensional “with memory”, while ordinary differential equations are finite-dimensional “memoryless” [5]. This makes it impossible to simply apply various mature analysis and synthesis methods in the classical control theory mainly for finite-dimensional systems in the analysis and synthesis of time-delay systems.
Time-delay systems and time-delay phenomena widely exist in system models such as mechanical, physical, chemical, and biological systems, economic systems, and population dynamic systems [6]. Especially in engineering application disciplines and various industrial practices, the time-delay phenomenon is a common phenomenon, such as signal delay in network control systems, congestion delay in high-speed communication networks, artificial neural network time delay, slow thermal reaction process, and long pipeline material transmission. [7] The existence of a large number of time-delay phenomena often leads to the instability of the whole system and the deterioration of the dynamic performance of the system. The complexity of the analysis of time-delay systems relative to nontime-delay systems and the difficulty of designing corresponding control systems make time-delay systems a research hotspot in the field of control theory and control engineering [8]. The stability analysis of various time-delay systems is the starting point and foundation of time-delay system research and has important theoretical research significance and engineering application value.
Stability analysis of time-delay systems is the starting point of time-delay system research and the foundation of time-delay system synthesis problems. The predecessors have obtained a large number of time-delay system stability criteria from various ideas and approaches [9]. These criteria can be divided into the time-delay-independent stability criterion and the time-delay-dependent stability criterion according to the dependence on delay time information. In general, the time-delay independence criterion is relatively simple, but the conclusions are conservative, especially for systems with only small delays, the time-delay-dependent stability criterion is less conservative, but the verification process is more complicated [10]. It is worth noting that “for some systems whose stability is not affected by time delay, no matter how much time-delay there is, the system can remain stable, but the time-delay dependence criterion is conservative [11]. Stability analysis of time-delay systems, like nontime-delay systems, can also be mainly divided into time-domain methods and frequency-domain methods. In the following, starting from these two methods, we will briefly review the development of stability analysis of time-delay systems, with emphasis on the results of the following years.
The development of neural networks (NNs) has a long history, which can be roughly divided into the following three stages: the first stage is the enlightenment period. Since 1940, scholars have devoted themselves to the study of neural networks. The second stage is the low tide period [12]. Due to the linear characteristics of the structure of the perceptron model and the limited functions, the perceptron model can only identify linear samples and cannot identify nonlinear samples; the third stage is the renaissance period. In 1982, American physicist Hopfield proposed the discrete Hopfield network, introduced the energy function into the neural network for the first time, and proved the stability of the network, which greatly promoted the development and research of neural networks [13]. Theoretically, the complexity of the model increases with the increase of parameters. The larger the capacity, the more complex the learning task can be accomplished by the model. On the contrary, the time and computational complexity of the complex model are high, and the training efficiency is relatively high.
At present, deep learning has been successfully applied in the fields of image recognition, speech recognition, natural language processing, and information retrieval [14]. Because of its wide potential market, a large number of scholars and enterprises are concerned about how to use deep learning to solve practical problems. For image classification and recognition, convolutional neural network (CNN) is an effective deep neural network method developed on the basis of neural networks. Since DBN ignores the two-dimensional structure of the image, some scholars combine DBN and CNN to propose Convolutional Deep Belief Networks (CDBN) by sharing weights at all positions in an image and obtaining the results [15].
The technical challenges faced by degree learning are mainly the problems faced by neural networks. As the number of layers continues to increase, the main challenges are: (1) there are few labeled datasets available, and the fine-tuning process of deep learning still uses supervised learning methods, so the quality of the classification model has a lot to do with the quantity and quality of the training dataset; (2) the interpretability of the network is low. Although deep learning can achieve good results and is widely used, it is still a “black box model” that is difficult to explain, because its internal knowledge expression is not intuitive, and the network learned is implied in the connection weights and biases, which is hard to understand. Some works have tried to improve the inexplicability of neural networks, mainly by extracting easy-to-understand symbolic rules from the knowledge representation of neural networks; and (3) overfitting. When the model learns the training samples are “too good” during the training process, it is likely to regard some characteristics of the training samples themselves as general properties of the potential samples, which leads to a reduction in the test error. It can also be said that the generalization ability is not strong, and this phenomenon is called overfitting [16, 17]. The strong learning ability and high complexity of neural networks may lead to overfitting. Data augmentation and regularization are two commonly used methods to solve overfitting.
Performance evaluation is an era requirement for higher education to develop from quantitative change to qualitative change [18]. For the Ministry of Education, it should consider how to effectively allocate resources among colleges and universities across the country to make the most of limited resources. Effective performance evaluation can promote the enthusiasm and autonomy of colleges and universities to improve work efficiency and innovate management methods [19]. At the same time, it can truthfully reflect the utilization of various resources in colleges and universities, the needs of colleges and universities, and the problems existing in the use of college resources, so as to guide colleges and universities to allocate corresponding resources. Therefore, as the first step in the optimal allocation of resources, college performance evaluation is increasingly urgent and important to enrich the evaluation system of colleges and universities.
This study is based on the research background of the project “Research on the Optimal Allocation of Assets and Resources in Colleges and Universities”. With the rapid development of higher education, how to allocate and manage scientific research resources in research-oriented colleges and universities is the most concerning issue for the Ministry of Education and colleges and universities. In the higher education circles at home and abroad, the input and output performance evaluation of human resources, financial resources, and material resources are regarded as relatively weak links with great challenges and risks. For the multi-input performance evaluation system, its goal is to find the mapping relationship between input, output characteristics, and performance evaluation, that is to say, the essence of performance evaluation in this study is to find a nonlinear relationship between multifeature modules and a single output variable. Linear function is to obtain a multiclass learning model. First, the stability of time-delay unified systems at zero and positive equilibrium is studied, and the conditions for generating Hopf branches are given. At the same time, some conclusions are obtained through theoretical analysis. Numerical simulations further verify the validity of the theoretical results, then apply the deep learning nonlinear model method to the input-output performance evaluation data of colleges and universities, and conduct comparative experiments with Softmax classifier, BP neural network, and deep belief network.
2. Methods and Experiment Design
2.1. Stability and Branching of Time-Delay Unified Systems
Since the discovery of the first chaotic system in 2000, many new chaotic systems have been discovered by various methods, such as the famous Rosse system. In 1999, Chen discovered a new system that is dual to the Lorenz system in the study of chaotic anti-control (or chaptalization) [20]. In 2002, Lu and Chen further discovered a new critical chaotic system, which was later called the Lu system. Soon after, they unified the Lorenz system, Lu system, and Chen system with a smooth convex transformation, called the unified system, which can be expressed as
We consider adding a time-delay term to the unified system to control its dynamical behavior, adding to the third item of the system:
When , the system has 3 balance points (0,0,0), (), (), where, , , when or , the system only has one balance point(0,0,0).
Linearizing the unified system at the origin can obtain the characteristic equation as
The discriminant of the second term quadratic polynomial of the characteristic equation is written as , then the root of equation (3) is
When , all roots have negative real parts origin in origin (0,0,0).
Similarly, we can obtain the root of the characteristic equation for (2)
Further simplification can be obtained:
When (8+ ) (6k-8- )<0, for all , the system equation (2) is all asymptotically stable.
Based on the above analysis, we have the following lemma:(1)If , for all , the characteristic root of the system (2) has a negative real part, and the system equation (2) is asymptotically stable(2)If (8+ ) (6k-8- )>0, for all , the characteristic root of the system equation (2) has a negative real part, and the system equation (2) is asymptotically stable
Using this lemma, we can get the following theorem about branching:
If is defined by (6), for all , the characteristic root of the system (2) has a negative real part, and the system equation (2) is asymptotically stable.
2.2. Stability and Branching of Positive Equilibrium
For the stability of the system in balance point:
Discussing the stability of the system (7) in balance point is equivalent to that in origin (0,0,0).
After linearization, we can get
Thus,
If , we can obtain that
Using the Hurwitz theorem, we can get the necessary and sufficient conditions for all roots of the equation (10) to have negative real parts are
Comprehensively available, when 0.4< <-0.0137, the system has a negative real part in .
When , we can obtain the phase diagrams as shown in Figures 1 and 2.
Lemma 1. Ifwhere is constant, thus when is changing, the sum of zero multiplicities of F on the half-plane changes if and only if a zero root appears on or crosses the imaginary axis.
Thus, we can obtain that: (1) if , the system is asymptotically stable; (2) , h(z) has at least a positive real root , and the original equation is asymptotically stable for ; (3) if the conditions in the (2) are all established, and , only when , the system produced Hopf branches in balance points.
2.3. Data Set and Its Preprocessing
The project of optimizing the allocation of assets and resources in colleges and universities aims to further promote the reform of asset resource allocation plans of colleges and universities and the construction of disciplines in colleges and universities, so as to reasonably allocate various resources of colleges and universities, and realize the sustainable development of social services, discipline construction, and personnel training in colleges and universities. It is convenient to lead schools to formulate strategies that are in line with the development of the discipline and the facts. The strategy and operation method of the optimal allocation of university resources proposed by the project not only meet the requirements of building an economical and economical society but also improve the utilization rate and fairness of educational investment resources. The resource optimization allocation strategy takes 72 research universities of the Ministry of Education as the main objects [21]. By analyzing the basic contradictions within the universities and the status quo of input and output, the optimization allocation is divided into four processes: performance evaluation, resource optimization allocation, risk assessment, and benefit prediction. That is to say, the initial performance evaluation is based on the input and output resources of colleges and universities, and the initial performance and expected performance are optimally allocated, and the risk of performance deviation is estimated, and the economic, academic and social benefits brought by the optimal allocation, and then Further improvements to the optimization configuration process.
The data set in this study is based on the input-output data of 72 colleges and universities collected from 2007 to 2011, and then the performance evaluation data of colleges and universities is obtained by comprehensively evaluating the performance of colleges and universities. The first-level indicators of input are composed of financial resources, human resources, material resources, and intangible resources, with 14 second-level index variables and 42 third-level index variables; the first-level indicators of output include personnel training, scientific research, social services, It consists of disciplinary level and social influence, with 16 secondary indicator variables and 55 tertiary indicator variables, with a total of 360 sample data.
The three-level indicator variables in the input-output performance evaluation data of colleges and universities are used as candidate features. Since some feature data are missing, or most of the data values are 0, or have little correlation with the performance evaluation indicators, this part of the indicator data is deleted [22]. Therefore, there are two-part feature modules consisting of 29 input features and 27 output features. The 29 input characteristics are: the total number of faculty members, the number of full-time teachers, the number of administrative staff, the number of teaching assistants, the number of experimental staff, the number of research and development staff, the proportion of full-time teachers with doctoral degrees, the proportion of full-time teachers above associate high school, 35 and/or 45-year-old teachers to full-time teachers, number of doctoral tutors, expert team, total value of fixed assets at the end of the year, school area, total school building area, teacher area, laboratory and self-study space area, total value of equipment, teaching and research,the total value of equipment, the number of paper books, the total expenditure, the total expenditure on education, the total expenditure on scientific research, the average expenditure on education per student, the number of national key disciplines, the number of national specialties, and the average value of comprehensive strength rankings (the first year), country orientation.
The 27 output features are the total number of students, the total number of graduates, the ratio of doctoral students, the ratio of master students, the number of 100 outstanding doctoral dissertations, the number of outstanding alumni, the number of academic or scientific (social science) works, the number of SSCI\SCIE \A&HCI combined papers, CSSCI\EI combined papers, CPCI-S papers, core journal papers, papers published in top 1/10 SCI journals with impact factor, citations of papers, the number of citations of SCI papers, the number of national natural science, scientific and technological progress and technological invention awards, the number of scientific research achievement awards (natural science and humanities and social sciences) of the Ministry of Education, the acceptance number of scientific and technological achievements, and the provincial natural science, technological progress, technological invention awards, the numbers, important international academic conferences, important domestic academic conferences, number of patents granted, income from scientific research, income from education, number of doctoral tutors (increment in the current year), expert teams (increment in the current year), number of national characteristic majors (increment in the current year), and comprehensive strength ranking Mean (increment for the year). Table 1 shows some sample data and characteristics in 2020.
The performance in the sample data is used as the output variable, and the original performance data are between 56 and 100. According to K-means clustering, the performance value is discretized into 4 category labels, which are expressed as 1, 2, 3, 4, and category labels. The larger the value, the better the performance, and the single output of the performance is discretized into a multioutput form. K-means clustering is still used to select training samples. The specific method is to perform cluster analysis on samples with performance category labels, K = 4, and select 300 samples near the center of the class as training samples, and the rest as test samples, as shown in Table 2.
Because the input features have different dimensions and dimensional units, the training samples and test samples need to be normalized separately. Two common methods of data normalization are min-max normalization and Z-score normalization. Let feature , is normalized features, the equation if normalization iswhere min() is the minimum value of , max() is the maximum value of , and std() is the standard deviation of .
3. Results and Discussion
3.1. Numerical Experiment
We assumed that ; thus, our system becomes the Lorenz system. We achieve the purpose of chaos control by adjusting the value of no and two, and the system is expressed as follows:
Obviously, the system has three balance points O(0,0,0), . Here, we only consider .
Therefore, we can obtain the characteristic equation of the system (15) in point:
When , the characteristic equation becomes
The three roots of the characteristic equation are 13.8546,0.0940 + 10.19451,0.09-10.19451. Thus, when , the system (15) is in chaos.
When . Assuming k = -1, have: . has two positive roots, and one negative root. .
Its numerical simulation diagram is shown in Figures 3–5.
(a)
(b)
(a)
(b)
(a)
(b)
(c)
(d)
It can be seen from the above simulation diagram that when , the system is partially stable, otherwise is not stable.
3.2. Analysis of Effectiveness of Deep Learning Models
The method in this study firstly uses a Boltzmann machine (RBMs) feature extractor to train input features and output feature data to obtain two parts of new feature data, then reconstruct the new features as the input features of Softmax classifier, and then combine the two-layer RBMs and the Softmax classifiers are connected to form a modular deep learning network for multiclassification. Finally, the BP algorithm is used to reverse fine-tune the entire network to find the optimal parameters and output the predicted performance evaluation results.
The prediction accuracy rate can reflect whether the algorithm effectively predicts the performance evaluation of colleges and universities. The prediction accuracy rate (Acc) is based on the comparison of the reconstruction errors of each layer of RBMs based on input and output feature data. The layers of the RBMs feature extractor for given input and output and the number is 3 (including the input layer), and after many experiments and comparisons, the number of nodes in each layer of input and output RBMs is finally determined to be (29, 14, 10) and (27, 17, 11) respectively. The final model structure is shown in Figure 6. Table 3 shows the simulation results of the algorithm in this study on the performance evaluation data of colleges and universities. The Softmax classifier in the algorithm is a model used alone for classification; the BP neural network is a multihidden layer neural network, the activation function is a sigmoid function, and the number of nodes in each layer is [56, 31, 21, 4]. DBN-Softmax is a deep belief network whose top-level classifier is a Softmax classifier, and the number of hidden layer nodes of two-layer RBM is (31, 21); on the basis of the method in this study, RBMs-Softmax is a classification model without fine-tuning, which is used to emphasize that reverse fine-tuning has a significant effect on improving the performance of the classifier.
According to Table 3, The following can be obtained: (1) when only the Softmax classifier is used for classification, with the increase in the number of iterations, the prediction accuracy rate is improved to a certain extent. When the number of iterations is set to 300, the accuracy rate is 78.3%. The accuracy rates of DBN-Softmax and the method in this study are both higher than Softmax, indicating that it is effective to extract features from the input data first and classify using new feature data; (2) the accuracy rate of the method in this study is higher than that of the BP neural network, which may be due to the BP neural network. The neural network is limited by the number of hidden layer nodes, a random selection of initial parameters, and no abstract feature extraction ability; (3) the prediction accuracy rate before fine-tuning the method in this study is 76.7%, and the accuracy rate after fine-tuning is increased by 11.6%. The importance of reverse fine-tuning to network parameter learning is shown, indicating that fine-tuning is effective in improving the performance of the classifier; (4) combined with the characteristics of the data in this study, the prediction accuracy rate of using two RBMs is higher than that of one RBM, namely DBN-Softmax. The feasibility and effectiveness of the model proposed in this study are verified.
3.3. Deep Learning Model Convergence Analysis
The network convergence is reflected in the fine-tuning process after combining multiple RBMs and Softmax classifiers. The algorithm efficiency is evaluated according to the convergence. The change curve of the loss function of equations (13) and (14) with the number of fine-tuning iterations is shown in Figure 7. Although the loss function value will fluctuate slightly, but in general, with the increase in the number of iterations, the loss function gradually decreases and tends to be stable, which means that the model in this study has a good convergence speed.
3.4. Analysis of the Output Results of the Deep Learning Model
We input 60 test samples into the trained network, obtain the new feature data of each sample data in two layers of RBMs, and classify through the final Softmax classifier to obtain the corresponding classification result of each sample. The input data of 5 samples (sample 1, sample 2, sample 3, sample 4, and sample 5) and the output values of RBMs in each layer of RBM are randomly selected as the data for the performance analysis of the RBMs feature extractor. The input data of input and output samples are complex, and the difference between samples and the separability between sample characteristics are not large. After a layer of RBM, the data distribution of each sample is clearer and the separability becomes stronger. After the two-layer RBM function, each sample presents a very clear data distribution, and the features have high separability. RBMs feature extractors can mine distributed feature representations of data, presenting them in a more clearly separable form. In short, the above shows that the model in this study obtains a more abstract feature representation for the input data through multilayer RBM, that is, the feature representation of RBM from the bottom to the high level becomes more and more abstract, the separability of the sample features becomes stronger, and there are fewer possible guesses. The more conducive to classification. The separability of the new features of the model in this study after passing through multiple RBMs is stronger than that of the DBN passing through a single RBM, which shows that the feature expression ability of the model in this study is higher than that of the DBN on the dataset of multifeature modules.
Table 4 shows the network output results of some test sample data in the Softmax classifier. The output results are divided into 4 categories. Each data in the table represents the probability value corresponding to each category, and the sum of the probability that each sample belongs to all 4 categories. If it is 1, in which category the sample has the largest probability value, the prediction performance evaluation of the sample is classified into which category. For example, in sample 1, the probability of belonging to category 1 is 0.94, which is greater than the probability of belonging to categories 2, 3, and 4, the predicted performance evaluation corresponding to sample 1 is recorded as 1.
The network output result of the test sample is changed to the predicted performance evaluation. Table 5 shows the confusion matrix M of the category labels of the method in this study. The category label of the row vector is the actual performance evaluation, and the category label of the column vector is the predicted performance evaluation of the network output. Precision and recall come from information retrieval and are used to measure the performance of search engines. This study uses precision and recall in the field of information retrieval to define the precision and recall of each category label as follows:where P(i) and R(i) are the precision and recall of the ith class, respectively. is the number of correct samples identified by the ith class label. are the ith column vector and the ith row vector of the confusion matrix M, respectively, and sum is the sum of the vectors.
The model in this study is evaluated from the average precision, average recall, and prediction accuracy. The average recall (AP) and average precision (AR) are shown in (19), where K represents the number of categories. The final experimental results are shown in Table 6. It can be seen from the table that the three evaluation indicators of the method in this study are all larger than BP neural network and DBN-Softmax, indicating that the method in this study is superior to other algorithms in the evaluation of input-output performance in colleges and universities, and is feasible and effective.
4. Conclusions
Deep learning is a major breakthrough in machine learning research. It aims to establish a deep network structure that can simulate the human brain for analysis and learning, interpret data through the mechanism of layer-by-layer abstract feature representation, and has excellent feature learning capabilities. Deep learning is a framework for building multilayer neural networks on unsupervised data proposed by Hinton et al. It uses a greedy layer-by-layer unsupervised method to solve the parameter optimization problem of deep neural networks.
In our study, a modular deep learning model based on RBMs is improved for the input and output feature module data of university performance evaluation. The model of this study is applied to the performance evaluation data of colleges and universities for experimental analysis. The results show that the prediction accuracy rate, average precision rate, and average recall rate of this method are 86.87%, 88.86%, and 88.33% respectively, which are higher than Softmax classifier, BP neural network, and DBN. In addition, after network fine-tuning, the accuracy of this method is increased by 11.6%, which reflects the importance of reverse fine-tuning for network parameter learning. From the convergence analysis of the network, it can be seen that the value of the loss function gradually decreases and tends to be stable with the increase of the number of iterations, indicating that the model in this study has a good convergence speed. The method in this study is compared with DBN. The results show that the new data features are more separable after the feature extraction of multiple RBMs, which shows that the method in this study has the ability to abstract feature expression layer by layer for the input data, and can extract features that are beneficial to classification or prediction. The above fully illustrates the effectiveness and superiority of the method in this study, and the use of the model to comprehensively evaluate the performance of colleges and universities under the influence of high-dimensional factors and obtain relatively satisfactory evaluation results.
In the research of a dynamical system, the stability of a time-delay unified system at zero equilibrium and positive equilibrium is studied, and the conditions for generating the Hopf branch are given. At the same time, some conclusions are obtained through theoretical analysis. Numerical simulations further verify the validity of the theoretical results.
In the future research study, we will first explore the relationship between the solutions of nonlinear differential equations, and use the solution method to find the transformation of nonlinear variable coefficient differential equations. Second, we will further study dynamical systems, study system stability, and the Hopf branch, and use the computer to carry out a numerical simulation to verify the correctness of the conclusion.
The model in this study has a higher prediction accuracy than other classification models and clarifies the effectiveness of the modular deep learning model based on RBMs from the perspectives of network convergence analysis and network output analysis. The feature expression ability of this model is stronger than DBN, the resulting abstract feature representation is more conducive to classification. Although the classification accuracy rate of the model in this study has been improved, the model has certain limitations. The network initialization is still set based on experiments and experience, and the prediction accuracy rate is only 88.3%, which needs to be improved. The parameter training algorithm of RBMs can be further studied to improve the more accurate reference basis for the performance evaluation of colleges and universities.
Data Availability
The experimental data used to support the findings of this study are available from the author upon request.
Conflicts of Interest
The author declares no conflicts of interest regarding the present study.