This article first introduces neural networks and their characteristics. Based on a comparison of the structure and function of biological neurons and artificial neurons, it focuses on the structure, classification, activation rules, and learning rules of neural network models. Based on the existing literature, this article adds a distributed time lag term of the neural network system. In the actual problem, history has a very important influence on the current change situation, and it is not only at a specific time in the past. It has an impact on the current state change rate. Therefore, based on the existing literature that only has discrete time lags, this paper adds distributed time lags. Such neural network systems can better reflect real-world problems. In this paper, we use three different inequality scaling methods to study the existence, uniqueness, and global asymptotic stability of a class of neural network systems with mixed delays and uncertain parameters. First, using the principle of homeomorphism, a new upper-norm norm is introduced for the correlation matrix of the neural network, and enough conditions for the existence of unique equilibrium points in several neural network systems are given. Under these conditions, the appropriate Lyapunov is used. Krasovskii functional, we prove that the equilibrium point of the neural network system is globally robust and stable. Numerical experiments show that the stability conditions of the neural network system we obtained are feasible, and the conservativeness of the stability conditions of the neural network system is reduced. Finally, some applications and problems of neural network models in psychology are briefly discussed.

#### Keywords

- Inequality scaling method
- Differential equation
- Artificial neural network
- Delay differential equation
- Psychology

#### MSC 2010

- 13J25

(1) Radical scaling:

(2) Enlarge or reduce the numerator or denominator in the fraction:

True fraction numerator and denominator decrease by a positive number at the same time, it becomes larger;

False fraction numerator and denominator decrease by a positive number at the same time, it becomes smaller:

(3) Apply basic inequality scaling:

(4) The binomial theorem shrinks:

(5) Round off (or add in) some items:

(6) First shrink the deflation term, then split it into the difference between adjacent terms of a certain sequence, and eliminate the middle term when summing. As shown in Figure 1.

Let the sum of the first _{n}

Let

Proof: easily available:

Comment: The key to this question is to split

(7) First shrink the deflation term, then split it into the sum of term

It is known that the sequence {_{n}_{n}_{1} = 2, _{n}_{n}_{n+1} − 1), _{n}_{n}_{n}_{n+1} > _{n}

Verification: when

From (I), we know that _{n}_{2n−1} ≥ _{2n−2} ≥ … ≥ _{2} and
_{1} = 1,

That is, when

The lag

Currently, the corresponding characteristic equations are

However, now the characteristic equation |_{ij} + _{ij}^{−λτ} − _{ij}_{ij} + _{ij}^{−λτ} + _{ij}^{−λτ} − _{ij}_{i}_{i}

The zero solution is progressively stable.

Regardless of whether it is |_{ij} + _{ij}^{−λτ} − _{ij}_{ij} + _{ij}^{−λτ} + _{ij}^{−λτ} − _{ij}^{z}

Let ^{z}^{r}t^{s}

If the polynomial

If the polynomial

And ^{′}^{′}

Regarding the determination of a function of the form F, such a problem that all its roots are real, can be solved according to the following two principles:

To make all the roots of function

Starting from a sufficiently large, ensure that there are no complex roots but only solid roots.

Definition 1: Let _{m,n} _{mn}^{m}t^{n}_{rs} ≠ 0 and the exponents _{rs}^{r}t^{s}_{mn}^{m}t^{n}_{mn} ≠ 0 is taken out in the above polynomial, then there is one of 1.

Zero distribution of ^{z}_{m,n} _{mn}^{m}t^{n}^{z}

The zero of function

To study the necessary and enough conditions for

Where
^{2} + ^{2} = 1, it can be assumed that
^{2} + ^{2}. If
^{2} + ^{2}, then ^{2} + ^{2} = 0 ⇒

This is true for all such terms in

Note that the prime minister in

If polynomial

For the case where the first item exists in

Among them,

Let’s prove that the function

Neural network is a model that simulates the human nervous system based on connectionist theory. It is a computer program with the ability to adapt, self-organize, and self-learn. The basic constituent units of a neural network are called nodes or units. The network system adjusts and changes the connection strength between neurons according to present rules, and implements adaptive, self-organizing, and self-learning in a parallel distributed processing (PDP) manner, thereby exhibiting wisdom like biological nervous systems. The process is shown in sub-figure a in Figure 2. Connectionism is the general term for the theoretical framework of parallel distributed knowledge representation and calculation of neural networks. It is the research theory of neural networks and their characteristics and construction of mental models. It is also called “neural computing” or “parallel distributed processing”. Semiotic-oriented cognitive psychology adopts the logical rules of explicit hierarchical arrangement to manipulate and process symbols in a serial manner, which is often called information processing psychology. Connectionist-oriented cognitive psychology is based on neurophysiology, integrates the cognitive functions and characteristics of the human brain, uses digital features instead of logical rules to transform information, and processes subsymbols in parallel. It is also an information processing theory in a broad sense. The process is shown in sub-figure b in Figure 2.

The connectionist mental model is based on neuroscience to simulate the human nervous system. It is more neurologically reasonable than the semiotic model. Simple units are linked together to have complex behaviours and abilities, and to a certain extent can work like the human brain. The simplification of the unit organization has produced many interesting characteristics, such as showing “content address” type memory, fuzzy or partial stimulus can extract memory traces in the entire network, etc. If a part of the network is damaged, it will also decline like the human brain, and memory performance will gradually decline with the degree of damage, rather than a one-time collapse. Damage to any part of a traditional symbolic system usually results in a catastrophic crash. Neural network models are good at things, and people are good at them, such as complex pattern recognition and fuzzy guessing. The neural network can also automatically generate new examples in the example training, and then form prototypes from the examples.

Based on foreign research, Chinese psychologists have used the connectionist neural network model to make useful attempts in the study of Chinese cognition, and have achieved certain results. Chen Ying and Peng Yiling applied the idea of parallel distributed processing to the study of Chinese cognitive simulation, and adopted a distributed storage structure and parallel processing operation process, and proposed the “Chinese character recognition and naming connectionism model” (CMRP). The model is a three-layer feedforward network: the input layer is a glyph representation layer consisting of 420 units, the hidden unit layer uses 200 units to realize the non-linear mapping conversion from glyphs to phonetic sounds, and the output layer is composed of 42 units Layer of characterization. Zhang Dongsong, Chen Yongming and Yu Bailinian proposed a neural network model (CRAM) for sentence lattice role assignment. The model uses a vocabulary distribution characterization input layer, two hidden cell layers and a lattice role output layer. The back-propagation algorithm is used. After adjusting the connection weights between the layers of the network, after 108 sentences of training, it achieved an accuracy of 87%, as shown sub-figure a in Figure 3. Ming Hong and Zhang Houzheng proposed a loose rule hybrid computing model (RPHM) for Chinese sentence reading. This model is a hybrid structural network combining parallel distributed processing and symbolism paradigm. It uses distributed and symbolic representations to coexist, and dynamic multi-source parallel interaction processing the mechanism is computerized in stages. The network establishes the information organization structure of each part of the system in the form of hierarchical representation, and constructs different types of interaction mechanisms between the various parts. The above models are some useful explorations made by Chinese psychologists in applying neural network models to Chinese cognitive research. The process is shown in sub-figure b in Figure 3.

Until now, there are many technical limitations in the construction of neural network models. First, a network with a small number of units can work well when dealing with small problems, but it runs when it encounters large problems that require hundreds of units to solve. Difficulties and errors, which are called “scaling problems”, are the fundamental limitation of neural networks. Similar problems also exist in the semiotic model. For example, an expert system works well in a small domain with limited rules, and fails when the domain is expanded and there are too many rules. Second, the network often requires a lot of training to complete a certain task, unlike the human brain, which can be learned after one or two trainings, which limits its application. Third, without a lot of guidance, the quality of the network’s work will be poor. The designer must carefully design the network parameters and make a lot of adjustments during the network operation. Even if the input stimulus is a specially prepared vector rather than many external stimuli processed by humans, most networks still need mentors and cannot achieve unsupervised self-organized learning.

Many researchers have criticized the value of neural network models as mental models. With the development of brain neuroscience, we have a deeper understanding of the structure and function of neurons. Biological neurons are not a simple biostable logic operation unit, but a super miniature biological information processing device. The brain is a giant super parallel neural network. The artificial neural network model does not work exactly like the brain, and its simulation of psychological phenomena and processes also has some problems:

Firstly, although the network model has some similarities with the neural network of the brain, it does not really have a rationality in a neurological sense. The real neuron is much more complicated. Its working state is not just inhibition and excitement, it also has many connection modes. Biological neurons can have more than 100,000 connections, but the network model is only connected to neurons in adjacent layers. The network model is not the same as the human brain in almost every respect.

Secondly, neural network learning is not the same as human learning. Although the Internet often makes mistakes made by people, it also often makes mistakes that humans do not make. People can often learn easily at one time, and can easily learn by adding and extending existing knowledge through migration and analogies, but neural networks cannot yet achieve knowledge transfer and analogy. The brain is a highly complex biological system that integrates a variety of dynamic special-purpose components. It is not as simple as interconnected with neural network models. Advanced cognitive activities are not easy to simulate through the web, and a symbolic rule system must be used. Ling and Marinow compared many well-known language learning networks with symbolic programs. After analysing the performance of all programs, it was found that neural network models cannot learn past-style English verbs, and programs that use regular methods do better.

Thirdly, the neural network model often only provides a special case of demonstration, it successfully completes some cognitive functions, but this does not mean that it is done in the way of the human brain, only that it can be done. It’s like the relationship between science and engineering. Engineering establishes a working system, and science tries to discover how existing systems work. The above criticism may be too harsh on a scientific field that is being developed and improved, but this is exactly the problem that connectionism will gradually solve. After a detailed analysis of the connectionist theoretical framework, Smolinsky pointed out that whether neural network models can complete advanced cognitive tasks and whether they can correctly simulate the brain remains to be scientifically tested, but neural networks can indeed be used between neural and cognitive Analysis of sub-symbol levels. In addition, further research is needed on the division of labour in cognitive research between neuroscience, connectionism, and semiotics. Under the theoretical framework of cognitive neuroscience, the combination of connectionist neural networks or neural computing and brain imaging will make a huge contribution to modern psychological science.

Neural networks are derived from researchers’ simulations of the computational capabilities of real neuron networks. During the development process, they gradually showed powerful cognitive functions such as learning, memory, and association. From the perspective of simulating real experiments, “the uncertainty factors in other methods can be solved here in the neural network model”; from the perspective of exploring the internal mechanism of the cognitive process, the parallel distributed processing method of the neural network It is as efficient and anti-interference as neuron’s information transmission, so it may be closer to the essence of cognition.

Law of interest rate changes in financial markets based on the differential equation model of liquidity Basalt fibre continuous reinforcement composite pavement reinforcement design based on finite element model Industrial transfer and regional economy coordination based on multiple regression model Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Spatial–temporal graph neural network based on node attention A contrastive study on the production of double vowels in Mandarin Research of cascade averaging control in hydraulic equilibrium regulation of heating pipe network Mathematical analysis of civil litigation and empirical research of corporate governance Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Performance evaluation of college laboratories based on fusion of decision tree and BP neural network Application and risk assessment of the energy performance contracting model in energy conservation of public buildings Sensitivity analysis of design parameters of envelope enclosure performance in the dry-hot and dry-cold areas The Spatial Form of Digital Nonlinear Landscape Architecture Design Based on Computer Big Data Analysis of the relationship between industrial agglomeration and regional economic growth based on the multi-objective optimisation model Constraint effect of enterprise productivity based on constrained form variational computing The impact of urban expansion in Beijing and Metropolitan Area urban heat Island from 1999 to 2019 TOPSIS missile target selection method supported by the posterior probability of target recognition Ultrasonic wave promoting ice melt in ice storage tank based on polynomial fitting calculation model The incentive contract of subject librarians in university library under the non-linear task importance Application of Fuzzy Mathematics Calculation in Quantitative Evaluation of Students’ Performance of Basketball Jump Shot Visual error correction of continuous aerobics action images based on graph difference function Application of Higher Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Application of Forced Modulation Function Mathematical Model in the Characteristic Research of Reflective Intensity Fibre Sensors Radioactive source search problem and optimisation model based on meta-heuristic algorithm Research on a method of completeness index based on complex model Fake online review recognition algorithm and optimisation research based on deep learning Research on the sustainable development and renewal of Macao inner harbour under the background of digitisation Support design of main retracement passage in fully mechanised coal mining face based on numerical simulation Study on the crushing mechanism and parameters of the two-flow crusher Interaction design of financial insurance products under the Era of AIoT Modeling the pathway of breast cancer in the Middle East Corporate social responsibility fulfilment, product-market competition and debt risk: Evidence from China ARMA analysis of the green innovation technology of core enterprises under the ecosystem – Time series data Reconstruction of multimodal aesthetic critical discourse analysis framework Image design and interaction technology based on Fourier inverse transform What does students’ experience of e-portfolios suggest Research on China interregional industrial transformation slowdown and influencing factors of industrial transformation based on numerical simulation The medical health venture capital network community structure, information dissemination and the cognitive proximity Data mining of Chain convenience stores location The optimal model of employment and entrepreneurship models in colleges and universities based on probability theory and statistics A generative design method of building layout generated by path Parameter Id of Metal Hi-pressure State Equation Analysis of the causes of the influence of the industrial economy on the social economy based on multiple linear regression equation Research of neural network for weld penetration control Intelligent Recommendation System for English Vocabulary Learning – Based on Crowdsensing Regarding new wave distributions of the non-linear integro-partial Ito differential and fifth-order integrable equations Research on predictive control of students’ performance in PE classes based on the mathematical model of multiple linear regression equation Beam control method for multi-array antennas based on improved genetic algorithm The influence of X fuzzy mathematical method on basketball tactics scoring Application of regression function model based on panel data in bank resource allocation financial risk management Research on aerobics training posture motion capture based on mathematical similarity matching statistical analysis Application of Sobolev-Volterra projection and finite element numerical analysis of integral differential equations in modern art design Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Research on motion capture of dance training pose based on statistical analysis of mathematical similarity matching Application of data mining in basketball statistics Application of B-theory for numerical method of functional differential equations in the analysis of fair value in financial accounting Badminton players’ trajectory under numerical calculation method Research on the influence of fuzzy mathematics simulation model in the development of Wushu market Study on audio-visual family restoration of children with mental disorders based on the mathematical model of fuzzy comprehensive evaluation of differential equation Difference-in-differences test for micro effect of technological finance cooperation pilot in China Application of multi-attribute decision-making methods based on normal random variables in supply chain risk management Exploration on the collaborative relationship between government, industry, and university from the perspective of collaborative innovation The impact of financial repression on manufacturing upgrade based on fractional Fourier transform and probability AtanK-A New SVM Kernel for Classification Validity and reliability analysis of the Chinese version of planned happenstance career inventory based on mathematical statistics Visual positioning system for marine industrial robot assembly based on complex variable function Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Research on the influencing factors of agricultural product purchase willingness in social e-commerce situation Study of a linear-physical-programming-based approach for web service selection under uncertain service quality A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China Calculating university education model based on finite element fractional differential equations and macro-control analysis Educational research on mathematics differential equation to simulate the model of children's mental health prevention and control system Analysis of enterprise management technology and innovation based on multilinear regression model Verifying the validity of the whole person model of mental health education activities in colleges based on differential equation RETRACTION NOTE Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Adoption of deep learning Markov model combined with copula function in portfolio risk measurement Good congruences on weakly U-abundant semigroups Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Mathematical simulation analysis of optimal detection of shot-putters’ best path Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Determination of the minimum distance between vibration source and fibre under existing optical vibration signals: a study Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets Nonlinear Differential Equations in the Teaching Model of Educational Informatisation Fed-UserPro: A user profile construction method based on federated learning The evaluation of college students’ innovation and entrepreneurship ability based on nonlinear model Smart Communities to Reduce Earthquake Damage: A Case Study in Xinheyuan, China Response Model of Teachers’ Psychological Education in Colleges and Universities Based on Nonlinear Finite Element Equations Institutional investor company social responsibility report and company performance Mathematical analysis of China's birth rate and research on the urgency of deepening the reform of art education First-principles calculations of magnetic and mechanical properties of Fe-based nanocrystalline alloy Fe _{80}Si_{10}Nb_{6}B_{2}Cu_{2}The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Has the belt and road initiative boosted the resident consumption in cities along the domestic route? – evidence from credit card consumption MCM of Student’s Physical Health Based on Mathematical Cone Attitude control for the rigid spacecraft with the improved extended state observer Sports health quantification method and system implementation based on multiple thermal physiology simulation Research on visual optimization design of machine–machine interface for mechanical industrial equipment based on nonlinear partial equations Research on identifying psychological health problems of college students by logistic regression model based on data mining Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching A mathematical model of PCNN for image fusion with non-sampled contourlet transform Nonlinear Differential Equations in Computer-Aided Modeling of Big Data Technology The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Cognitive Computational Model Using Machine Learning Algorithm in Artificial Intelligence Environment Application of Higher-Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Informatisation of educational reform based on fractional differential equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis Research on the control of quantitative economic management variables under the numerical method based on stochastic ordinary differential equations Network monitoring and processing accuracy of big data acquisition based on mathematical model of fractional differential equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based on Fractional Differential Equations Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression System dynamics model of output of ball mill Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Mathematical function data model analysis and synthesis system based on short-term human movement Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Human gait modelling and tracking based on motion functionalisation Analysis and synthesis of function data of human movement The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Design of Morlet wavelet neural network to solve the non-linear influenza disease system Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate Differential equation model of financial market stability based on Internet big data 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Children’s cognitive function and mental health based on finite element nonlinear mathematical model Motion about equilibrium points in the Jupiter-Europa system with oblateness Fractional Differential Equations in Electronic Information Models Badminton players’ trajectory under numerical calculation method BIM Engineering Management Oriented to Curve Equation Model Optimal preview repetitive control for impulse-free continuous-time descriptor systems Development of main functional modules for MVB and its application in rail transit Study on the impact of forest fire prevention policy on the health of forest resources Mathematical Method to Construct the Linear Programming of Football Training The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Value Creation of Real Estate Company Spin-off Property Service Company Listing Selection by differential mortality rates Digital model creation and image meticulous processing based on variational partial differential equation Dichotomy model based on the finite element differential equation in the educational informatisation teaching reform model Nonlinear Dissipative System Mathematical Equations in the Multi-regression Model of Information-based Teaching The modelling and implementation of the virtual 3D animation scene based on the geometric centre-of-mass algorithm The policy efficiency evaluation of the Beijing–Tianjin–Hebei regional government guidance fund based on the entropy method The transfer of stylised artistic images in eye movement experiments based on fuzzy differential equations Research on behavioural differences in the processing of tenant listing information: An eye-movement experiment A review of the treatment techniques of VOC Some classes of complete permutation polynomials in the form of ( x ^{pm}−x +δ )^{s}+ax ^{pm}+bx overF _{p2m}The consistency method of linguistic information and other four preference information in group decision-making Research on the willingness of Forest Land’s Management Rights transfer under the Beijing Forestry Development A mathematical model of the fractional differential method for structural design dynamics simulation of lower limb force movement step structure based on Sanda movement Fractal structure of magnetic island in tokamak plasma Numerical calculation and study of differential equations of muscle movement velocity based on martial articulation body ligament tension Study on the maximum value of flight distance based on the fractional differential equation for calculating the best path of shot put Sports intensity and energy consumption based on fractional linear regression equation Analysis of the properties of matrix rank and the relationship between matrix rank and matrix operations Study on Establishment and Improvement Strategy of Aviation Equipment Research on Financial Risk Early Warning of Listed Companies Based on Stochastic Effect Mode Characteristics of Mathematical Statistics Model of Student Emotion in College Physical Education Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Nonlinear strategic human resource management based on organisational mathematical model Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Fractional Differential Equations in Sports Training in Universities Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model