Mathematical modeling teaching learning system

Abstract

The invention discloses a mathematical modeling teaching learning system, which relates to the field of mathematical learning and comprises a knowledge preparation module, a mathematical case library, an understanding analysis module, a selection construction module, a modeling evaluation module, a simulation verification module, an evaluation improvement module, a solution optimization module, an analysis verification module and a knowledge recommendation module. According to the method, the understanding and practical ability of students on optimization problems can be improved, the students are helped to understand the iterative process and convergence of an algorithm, the solving ability of the students in multi-objective optimization is enhanced, real-time feedback and result analysis can be provided, and the learning efficiency is improved; more accurate feedback can be provided, students are helped to identify and correct errors, the depth of modeling thinking is improved, the understanding of the students on the decision process is enhanced, the efficiency and accuracy of model selection are improved, model convergence is accelerated, and computing resources are saved.

Description

Technical Field

[0001] The present invention relates to the field of mathematics learning, and particularly to a mathematical modeling teaching and learning system. Background Art

[0002] Mathematical modeling is a process of transforming practical problems into mathematical language and mathematical models, solving them through mathematical methods, and ultimately providing support for decision-making. It is an indispensable part of modern science and engineering technology and is closely related to disciplines such as computer science, statistics, and operations research. With the increasing demand of society for the ability to solve complex problems, the importance of mathematical modeling education has become increasingly prominent. However, traditional mathematical modeling teaching often focuses on the imparting of theoretical knowledge and ignores the cultivation of students' practical abilities. Traditional mathematical modeling teaching models face problems such as boring teaching content, single learning methods, and lack of effective practical platforms. This makes many students lack sufficient confidence and ability to effectively model and solve practical problems when facing them. How to improve the quality of mathematical modeling teaching and help students better understand and master modeling skills has become a key issue in current education reform.

[0003] After retrieval, Chinese Patent No. CN113935192A discloses a mathematical modeling teaching and learning system. Although this invention can significantly improve the learning efficiency of mathematical modeling learning, it cannot improve students' understanding and practical abilities of optimization problems, is not conducive to helping students understand the iterative process and convergence of algorithms, and reduces students' problem-solving abilities in multi-objective optimization. In addition, existing mathematical modeling teaching and learning systems cannot provide more accurate feedback, reduce the depth of modeling thinking, and have low efficiency and accuracy in model selection. For this reason, we propose a mathematical modeling teaching and learning system. Summary of the Invention

[0004] The object of the present invention is to address the above-mentioned problems and provide a mathematical modeling teaching and learning system.

[0005] The present invention proposes a mathematical modeling teaching and learning system, which includes: a knowledge preparation module, a mathematical case library, an understanding and analysis module, a selection and construction module, a modeling evaluation module, a simulation verification module, an evaluation and improvement module, a solution optimization module, an analysis and verification module, and a knowledge recommendation module;

[0006] The knowledge preparation module is used to build a mathematical knowledge network based on the basic knowledge of mathematics and computer science required for mathematical modeling;

[0007] The mathematical case library is used to collect various classic mathematical modeling cases covering different fields and difficulty levels;

[0008] The understanding and analysis module is used to guide students to understand the background, objectives, and constraints of the problem, identify the key factors and variables of the problem, and establish a preliminary model framework;

[0009] The selection and construction module selects the corresponding mathematical model according to the characteristics of the problem;

[0010] The simulation and verification module is used to transform the established mathematical model into a computer simulation model;

[0011] The evaluation and improvement module is used to evaluate the quality of the model established by students and put forward improvement suggestions;

[0012] The solution and optimization module is used to explore and optimize various solution algorithms based on the established mathematical model;

[0013] The analysis and verification module is used to help students analyze the solution results;

[0014] The modeling evaluation module is used to search for the optimal modeling scheme to judge the modeling process and results of students;

[0015] The knowledge recommendation module is used to recommend personalized learning resources by combining students' learning history, knowledge mastery, and modeling cases.

[0016] As a further solution of the present invention, the specific steps for the knowledge preparation module to construct a mathematical knowledge network are as follows:

[0017] S1.1: Collect the text data of textbooks, lecture notes, and online resources, preprocess each group of collected text data, perform sentence splitting and word segmentation on each text data, and then use natural language processing technology to identify the entities in the text, and use the identified entities as nodes;

[0018] S1.2: Define the meaning, attributes, and corresponding descriptions of each identified entity, analyze the relationships between entities in the text through pattern matching methods, and use the identified relationships as edges to connect the corresponding entity nodes to establish a preliminary mathematical knowledge network;

[0019] S1.3: Traverse the preliminary mathematical knowledge network, merge the same entities from different sources, eliminate the ambiguity of entity names according to the context analysis in the text data, store the processed mathematical knowledge network through the Neo4j database, and establish indexes for nodes and relationships.

[0020] As a further solution of the present invention, the specific steps for the simulation and verification module to transform the established mathematical model into a computer simulation model are as follows:

[0021] S2.1: Select the corresponding discretization method according to the characteristics of the mathematical model selected by the selection construction module, set the time step, convert the continuous mathematical model into a discrete difference equation, and then select the simulation software according to the characteristics and requirements of the mathematical model;

[0022] S2.2: Build a model in the simulation software, define variables, parameters, equations and various model parameters of the module, set the initial values, boundary conditions and physical parameter values of the simulation model, unify the units of each parameter, and then set the initial state of the mathematical model;

[0023] S2.3: Load and run the corresponding simulation program, simulate the operation process of the mathematical model, collect the simulation data of state variables, output variables and intermediate variables in real time, then store the simulation data in a file, and at the same time visually display the simulation data.

[0024] As a further solution of the present invention, the specific steps for the solution optimization module to explore and optimize each solution algorithm are as follows:

[0025] S3.1: Based on the solver parameter range, solution algorithm and combination methods of different solution algorithms of the current mathematical model, construct a solution space including a vector of solver parameter values and a sequence of multiple groups of solution algorithm combination methods, and take each solution algorithm parameter as a different node in the solution space, and initialize a group of populations;

[0026] S3.2: Initialize the node positions of each individual in the population in the solution space, initialize the pheromone values of each node in the solution space to a unified value, then take the efficiency of each solution algorithm as the heuristic information of each node, and each individual calculates the transition probability of the next parameter node based on the pheromone value and heuristic information of the current node, and randomly selects a node to move based on the selection probability of each node;

[0027] S3.3: Gradually perform node selection until a complete path, that is, a complete solution strategy, is constructed, and calculate the fitness value of this solution strategy based on the solution accuracy and efficiency. Then repeat the node selection and path construction until the change value of the fitness values of the paths constructed by each group of individuals converges to the preset range, and then stop the iteration;

[0028] S3.4: Compare the fitness values of the paths constructed by each group of individuals, and select the solution strategy corresponding to the path with the highest fitness value as the optimal strategy, and apply it to the actual solution process of the mathematical model at the same time.

[0029] As a further solution of the present invention, the specific steps for the evaluation and improvement module to evaluate the quality of the model established by the student are as follows:

[0030] S4.1: Collect a large number of mathematical modeling cases submitted by students, extract the feature data of model type, model parameters, and code quality from the students' modeling solutions, divide the dataset into a training set, a validation set, and a test set, establish an evaluation model based on the FCNN model architecture, and initialize the evaluation model parameters;

[0031] S4.2: Input the training set into the evaluation model. The input layer of the model receives each training set data, performs forward propagation on each group of training set data to obtain the corresponding predicted values, calculates the loss value between the predicted values and the actual values through the cross-entropy loss function, and then performs backpropagation of the loss value starting from the output layer of the evaluation model based on the chain rule, and adjusts the weights and biases of the model;

[0032] S4.3: Use the validation set to evaluate the performance of the model, adjust the hyperparameters of the model based on the evaluation performance, and then use the test set to evaluate the performance of the model. If the model performance reaches the preset indicators, deploy the trained model to the system; otherwise, repeatedly perform training, validation, and testing until the loss value of the evaluation model converges within the preset threshold;

[0033] S4.4: Input the current mathematical modeling data into the evaluation model. The evaluation model outputs the quality of the mathematical model established by the current student through forward propagation and proposes improvement suggestions.

[0034] As a further solution of the present invention, the specific steps for the modeling evaluation module to search for the optimal modeling solution to judge the students' modeling process and results are as follows:

[0035] S5.1: Collect the selected model type, selected feature variables, adjusted model parameters, and final model evaluation indicators, establish a model state space, and then based on the optional operations of selecting a new model type, adding or deleting feature variables, and adjusting model parameters, establish a corresponding action space. Create a root node based on the current model state with the selected operation, generate multiple groups of child nodes according to the optional operations of the root node, and each child node represents a model state to generate an evaluation tree;

[0036] S5.2: Starting from the root node, calculate the upper confidence bound value of each child node, and layer by layer select the child node with the highest upper confidence bound value until an unexplored child node is reached. If the reached node is not a terminal node, expand one or more child nodes from this child node;

[0037] S5.3: Starting from the expanded child nodes, perform random simulation until a terminal node is reached, that is, complete a modeling solution, and backpropagate the simulation results to all the nodes passed through, update the access times and reward values of each node, and repeatedly perform selection, expansion, simulation, and backtracking until the preset number of iterations is reached;

[0038] S5.4: After the iteration ends, traverse the evaluation tree and select the child node of the root node with the most access times as the optimal modeling solution. Compare the student's modeling solution with the found optimal solution, evaluate the advantages and disadvantages of the student's modeling process based on the model performance indicators, the rationality and innovation of the modeling process, and provide personalized feedback and suggestions to the student according to the evaluation results.

[0039] Advantages of the present invention:

[0040] 1. According to the solver parameter range, solution algorithms, and combination methods of different solution algorithms of the current mathematical model, the present invention constructs a solution space including a vector of solver parameter values and a sequence of multiple groups of solution algorithm combination methods, takes each solution algorithm parameter as a different node in the solution space, initializes a group of populations, initializes the node positions of each individual in the solution space in the solution space, and initializes the pheromone values of each node in the solution space to a unified value. Then, takes the efficiency of each solution algorithm as the heuristic information of each node. Each individual calculates the transition probability of the next parameter node based on the pheromone value and heuristic information of the current node, and randomly selects a node to move based on the selection probability of each node, and gradually selects nodes until a complete path, that is, a complete solution strategy, is constructed. Calculate the fitness value of this solution strategy based on the solution accuracy and efficiency. Then, repeat the node selection and path construction until the change value of the fitness values of the paths constructed by each group of individuals converges to the preset range, stop the iteration, compare the fitness values of the paths constructed by each group of individuals, and select the solution strategy corresponding to the path with the highest fitness value as the optimal strategy, and at the same time apply it to the actual mathematical model solution process, which can improve the student's understanding and practical ability of optimization problems, help the student understand the iterative process and convergence of the algorithm, enhance the student's problem-solving ability in multi-objective optimization, can provide real-time feedback and result analysis, and improve the learning efficiency.

[0041] 2. The present invention collects the selected model type, selected feature variables, adjusted model parameters, and final model evaluation metrics, and establishes a model state space. Then, based on the optional operations of selecting a new model type, adding or deleting feature variables, and adjusting model parameters, a corresponding action space is established. A root node is created based on the selected operation for the current model state, and multiple sets of child nodes are generated according to the optional operations of the root node. Each child node represents a model state to generate an evaluation tree. Starting from the root node, the upper confidence bound value of each child node is calculated, and the child node with the highest upper confidence bound value is selected layer by layer until an unexplored child node is reached. If the reached node is not a termination node, one or more child nodes are expanded from this child node. Starting from the expanded child nodes, random simulation is performed until a termination node is reached, which completes a modeling scheme. Then, the simulation results are backpropagated to all the nodes passed through, updating the visit times and reward values of each node. The processes of selection, expansion, simulation, and backtracking are repeated until the preset number of iterations is reached. After the iteration ends, the evaluation tree is traversed, and the child node of the root node with the most visit times is selected as the optimal modeling scheme. The modeling scheme of the student is compared with the found optimal scheme, and the advantages and disadvantages of the student's modeling process are evaluated based on the model performance metrics, rationality, and innovation of the modeling process. According to the evaluation results, personalized feedback and suggestions are provided to the student, which can provide more accurate feedback, help the student identify and correct errors, enhance the depth of the modeling thinking, strengthen the student's understanding of the decision-making process, improve the efficiency and accuracy of model selection, accelerate model convergence, and save computing resources. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] The present invention will be further described below with reference to the accompanying drawings.

[0043] Figure 1 It is a framework diagram of a mathematical modeling teaching and learning system. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0044] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0045] All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0046] Embodiment 1

[0047] The embodiment of the present invention provides a mathematical modeling teaching and learning system. SeeFigure 1 , Figure 1 This is a framework diagram of a mathematical modeling teaching and learning system provided by an embodiment of the present invention. The system includes: a knowledge preparation module, a mathematical case library, an understanding and analysis module, a selection and construction module, a modeling evaluation module, a simulation verification module, an evaluation and improvement module, a solution optimization module, an analysis and verification module, and a knowledge recommendation module.

[0048] The knowledge preparation module is used to build a mathematical knowledge network according to the basic knowledge of mathematics and computer science required for mathematical modeling.

[0049] Specifically, collect various text data of textbooks, lecture notes, and online resources, preprocess the collected groups of text data, perform sentence splitting and word segmentation on each text data, then use natural language processing technology to identify entities in the text, use the identified entities as nodes, define the meaning, attributes, and corresponding descriptions of each identified entity, analyze the relationships between entities in the text through pattern matching methods, and use the identified relationships as edges to connect the corresponding entity nodes to establish a preliminary mathematical knowledge network. Traverse the preliminary mathematical knowledge network, merge the same entities from different sources, and eliminate the ambiguity of entity names according to the context analysis in the text data. Store the processed mathematical knowledge network in a Neo4j database and establish indexes for nodes and relationships.

[0050] The mathematical case library is used to collect various classic mathematical modeling cases covering different fields and difficulty levels.

[0051] The understanding and analysis module is used to guide students to understand the background, objectives, and constraints of the problem, identify the key factors and variables of the problem, and establish a preliminary model framework.

[0052] The selection and construction module selects the corresponding mathematical model according to the characteristics of the problem.

[0053] The simulation verification module is used to convert the established mathematical model into a computer simulation model.

[0054] Specifically, select the corresponding discretization method according to the characteristics of the mathematical model selected by the selection and construction module, set the time step, convert the continuous mathematical model into a discrete difference equation, then select the simulation software according to the characteristics and requirements of the mathematical model, build the model in the simulation software, and define the model parameters of variables, parameters, equations, and modules, set the initial values, boundary conditions, and physical parameters of the simulation model, unify the units of all parameters, then set the initial state of the mathematical model, load and run the corresponding simulation program, simulate the operation process of the mathematical model, collect the simulation data of state variables, output variables, and intermediate variables in real time, then store the simulation data in a file, and at the same time visually display the simulation data.

[0055] The evaluation and improvement module is used to evaluate the quality of the model established by students and propose improvement suggestions.

[0056] Specifically, collect a large number of mathematical modeling cases submitted by students, extract various characteristic data such as model type, model parameters, and code quality from the students' modeling solutions, divide the data set into a training set, a validation set, and a test set, establish an evaluation model based on the FCNN model architecture, and initialize the evaluation model parameters. Input the training set into the evaluation model. The input layer of the model receives each training set data and performs forward propagation on each group of training set data to obtain corresponding predicted values. Calculate the loss value between the predicted value and the actual value through the cross-entropy loss function. Then, starting from the output layer of the evaluation model, perform backpropagation based on the chain rule and adjust the weights and biases of the model. Use the validation set to evaluate the performance of the model, and adjust the hyperparameters of the model based on the evaluation performance. Then use the test set to evaluate the performance of the model. If the model performance reaches the preset index, deploy the trained model to the system. Otherwise, repeatedly perform training, validation, and testing until the loss value of the evaluation model converges within the preset threshold. Input the current mathematical modeling data value into the evaluation model. The evaluation model outputs the quality of the mathematical model established by the current student through forward propagation and proposes improvement suggestions.

[0057] Embodiment 2

[0058] The embodiment of the present invention provides a mathematical modeling teaching and learning system. Refer to Figure 1 , Figure 1 which is a framework diagram of a mathematical modeling teaching and learning system provided by the embodiment of the present invention. The system includes: a knowledge preparation module, a mathematical case library, an understanding and analysis module, a selection and construction module, a modeling evaluation module, a simulation verification module, an evaluation and improvement module, a solution optimization module, an analysis and verification module, and a knowledge recommendation module.

[0059] The solution optimization module is used to explore and optimize each solution algorithm based on the established mathematical model.

[0060] Specifically, according to the solver parameter range, solution algorithm, and combination methods of different solution algorithms of the current mathematical model, a solution space is constructed that includes a vector of solver parameter values and a sequence of multiple groups of solution algorithm combination methods. Each solution algorithm parameter is regarded as a different node in the solution space. Initialize a group of populations, initialize the node positions of each individual in the solution space in the solution space, and initialize the pheromone values of each node in the solution space to a unified value. Then, use the efficiency of each solution algorithm as the heuristic information of each node. Each individual calculates the transition probability of the next parameter node based on the pheromone value and heuristic information of the current node, and randomly selects a node to move based on the selection probability of each node, gradually making node selections until a complete path, that is, a complete solution strategy, is constructed. Calculate the fitness value of this solution strategy based on the solution accuracy and efficiency. Then, repeat the node selection and path construction until the change value of the fitness values of the paths constructed by each group of individuals converges to the preset range, and then stop the iteration. Compare the fitness values of the paths constructed by each group of individuals, and select the solution strategy corresponding to the path with the highest fitness value as the optimal strategy, and apply it to the actual mathematical model solution process at the same time.

[0061] The analysis and verification module is used to help students analyze the solution results.

[0062] The modeling evaluation module is used to search for the optimal modeling scheme to evaluate the student's modeling process and results.

[0063] Specifically, collect the selected model type, selected feature variables, adjusted model parameters, and final model evaluation indicators, and establish a model state space. Then, based on the optional operations of selecting a new model type, adding or deleting feature variables, and adjusting model parameters, establish the corresponding action space. Create a root node based on the current model state and the selected operations. Generate multiple groups of child nodes according to the optional operations of the root node. Each child node represents a model state to generate an evaluation tree. Starting from the root node, calculate the upper confidence bound value of each child node, and layer by layer select the child node with the highest upper confidence bound value until an unexplored child node is reached. If the reached node is not a terminal node, expand one or more child nodes from this child node. Starting from the expanded child nodes, perform random simulations until a terminal node is reached, that is, complete a modeling scheme, and backpropagate the simulation results to all the nodes passed through, update the visit times and reward values of each node, and repeatedly perform selection, expansion, simulation, and backtracking until the preset number of iterations is reached. After the iteration ends, traverse the evaluation tree, and select the child node of the root node with the most visit times as the optimal modeling scheme. Compare the student's modeling scheme with the found optimal scheme, evaluate the pros and cons of the student's modeling process based on the model performance indicators, the rationality and innovation of the modeling process, and provide personalized feedback and suggestions to the student according to the evaluation results.

[0064] The knowledge recommendation module is used to recommend personalized learning resources by combining the learning history of students, their knowledge mastery, and modeling cases.

[0065] The above has described an embodiment of the present invention in detail, but the content described is only a preferred embodiment of the present invention and cannot be considered as limiting the scope of implementation of the present invention. All equivalent changes and improvements made according to the scope of the application of the present invention shall still fall within the scope covered by the patent of the present invention.

Claims

1. A mathematical modeling teaching and learning system, characterized in that, Including: A knowledge preparation module, a mathematical case library, an understanding and analysis module, a selection and construction module, a modeling evaluation module, a simulation verification module, an evaluation and improvement module, a solution optimization module, an analysis and verification module, and a knowledge recommendation module; The knowledge preparation module is used to build a mathematical knowledge network based on the basic knowledge of mathematics and computer science required for mathematical modeling; The mathematical case library is used to collect various classic mathematical modeling cases covering different fields and difficulty levels; The understanding and analysis module is used to guide students to understand the background, objectives, and constraints of the problem, identify the key factors and variables of the problem, and establish a preliminary model framework; The selection and construction module selects the corresponding mathematical model according to the characteristics of the problem; The simulation verification module is used to convert the established mathematical model into a computer simulation model; The evaluation and improvement module is used to evaluate the quality of the model established by students and put forward improvement suggestions; The solution optimization module is used to explore and optimize various solution algorithms based on the established mathematical model; The analysis and verification module is used to help students analyze the solution results; The modeling evaluation module is used to search for the optimal modeling scheme to judge the modeling process and results of students; The knowledge recommendation module is used to recommend personalized learning resources by combining the learning history, knowledge mastery, and modeling cases of students.

2. The mathematical modeling teaching and learning system according to claim 1, characterized in that, The specific steps for the knowledge preparation module to build a mathematical knowledge network are as follows: S1.1: Collect the text data of textbooks, lecture notes, and online resources, preprocess each group of collected text data, perform sentence splitting and word segmentation on each text data, and then use natural language processing technology to identify the entities in the text. Take each identified entity as a node; S1.2: Define the meaning, attributes, and corresponding descriptions of each identified entity, analyze the relationships between entities in the text through pattern matching methods, and take the identified relationships as edges to connect the corresponding entity nodes to establish a preliminary mathematical knowledge network; S1.3: Traverse the preliminary mathematical knowledge network, merge the same entities from different sources, eliminate the ambiguity of entity names according to the context analysis in the text data, store the processed mathematical knowledge network through the Neo4j database, and establish indexes for nodes and relationships.

3. The mathematical modeling teaching and learning system according to claim 2, wherein The specific steps for the simulation verification module to convert the established mathematical model into a computer simulation model are as follows: S2.1: Select the corresponding discretization method according to the characteristics of the mathematical model selected by the selection and construction module, set the time step, convert the continuous mathematical model into a discrete difference equation, and then select the simulation software according to the characteristics and requirements of the mathematical model; S2.2: Build a model in the simulation software, define variables, parameters, equations, and various model parameters of the module, set the initial values, boundary conditions, and physical parameters of the simulation model, unify the units of each parameter, and then set the initial state of the mathematical model; S2.3: Load and run the corresponding simulation program, simulate the running process of the mathematical model, collect various simulation data of state variables, output variables, and intermediate variables in real time, store the simulation data in a file, and at the same time visually display the simulation data.

4. The mathematical modeling teaching and learning system according to claim 3, characterized in that The specific steps for the solution optimization module to explore and optimize each solution algorithm are as follows: S3.1: According to the solver parameter range, solution algorithm, and combination methods of different solution algorithms of the current mathematical model, construct a solution space that includes a vector of solver parameter values and a sequence of multiple groups of solution algorithm combination methods. Take each solution algorithm parameter as a different node in the solution space, and initialize a group of populations; S3.2: Initialize the node positions of each individual in the population in the solution space, and initialize the pheromone value of each node in the solution space to a unified value. Then, take the efficiency of each solution algorithm as the heuristic information of each node. Each individual calculates the transition probability of the next parameter node based on the pheromone value and heuristic information of the current node, and randomly selects a node to move based on the selection probability of each node; S3.3: Gradually perform node selection until a complete path, that is, a complete solution strategy, is constructed. Calculate the fitness value of this solution strategy based on solution accuracy and efficiency. Then, repeat node selection and path construction until the change value of the fitness value of the paths constructed by each group of individuals converges to the preset range, and then stop the iteration; S3.4: Compare the fitness values of the paths constructed by each group of individuals, and select the solution strategy corresponding to the path with the highest fitness value as the optimal strategy, and apply it to the actual mathematical model solution process.

5. The mathematical modeling teaching and learning system according to claim 4, wherein The specific steps for the evaluation and improvement module to evaluate the quality of the model established by students are as follows: S4.1: Collect a large number of mathematical modeling cases submitted by students, extract various characteristic data such as model type, model parameters, model parameters, and code quality from the students' modeling solutions. Then, divide the data set into a training set, a validation set, and a test set. Establish an evaluation model based on the FCNN model architecture and initialize the evaluation model parameters; S4.2: Input the training set into the evaluation model. The input layer of the model receives each training set data and performs forward propagation on each group of training set data to obtain the corresponding predicted values. Calculate the loss value between the predicted values and the actual values through the cross-entropy loss function. Then, start from the output layer of the evaluation model, perform backpropagation based on the chain rule, and adjust the weights and biases of the model; S4.3: Use the validation set to evaluate the performance of the model, and adjust the hyperparameters of the model based on the evaluation performance. Then, use the test set to evaluate the performance of the model. If the model performance reaches the preset index, deploy the trained model to the system. Otherwise, repeatedly perform training, validation, and testing until the loss value of the evaluation model converges within the preset threshold; S4.4: Input the current mathematical modeling data into the evaluation model. The evaluation model outputs the quality of the mathematical model established by the current student through forward propagation and proposes improvement suggestions.

6. The mathematical modeling teaching and learning system according to claim 1, characterized in that The specific steps for the modeling evaluation module to search for the optimal modeling solution to judge the modeling process and results of students are as follows: S5.1: Collect the selected model type, selected feature variables, adjusted model parameters, and final model evaluation metrics, establish a model state space, and then based on the optional operations of selecting a new model type, adding or deleting feature variables, and adjusting model parameters, establish the corresponding action space. Create a root node based on the selected operation in the current model state, generate multiple sets of child nodes according to the optional operations of the root node, and each child node represents a model state to generate an evaluation tree; S5.2: Starting from the root node, calculate the upper confidence bound values of each child node, and layer by layer select the child node with the highest upper confidence bound value until an unexplored child node is reached. If the reached node is not a terminal node, expand one or more child nodes from this child node; S5.3: Starting from the expanded child nodes, perform random simulations until a terminal node is reached, that is, complete a modeling scheme, and backpropagate the simulation results to all the nodes passed through, update the visit times and reward values of each node, and repeatedly perform selection, expansion, simulation, and backtracking until the preset number of iterations is reached; S5.4: After the iteration ends, traverse the evaluation tree, and select the child node of the root node with the most visit times as the optimal modeling scheme. Compare the student's modeling scheme with the found optimal scheme, evaluate the pros and cons of the student's modeling process based on the model performance indicators, the rationality and innovation of the modeling process, and provide personalized feedback and suggestions to the student according to the evaluation results.

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