Symbolic regression method based on breeding elitist pool and population size diversity genetic programming
By optimizing the symbolic regression model using a genetic programming approach based on breeding elite pools and population size diversity, the inefficiency of traditional methods in complex problems is solved. This approach achieves a symbolic regression model with high accuracy and low complexity within a limited time, improving the interpretability and computational efficiency of the model.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- CHONGQING UNIV OF POSTS & TELECOMM
- Filing Date
- 2024-05-24
- Publication Date
- 2026-07-21
AI Technical Summary
Traditional genetic programming methods are inefficient in searching complex problems, making it difficult to obtain a symbolic regression solution with reasonable accuracy and acceptable complexity within a limited time. Furthermore, multi-objective genetic programming algorithms are prone to evolutionary degradation, resulting in algorithm inefficiency.
We employ a genetic programming approach based on breeding elite pools and population size diversity. This approach involves initializing the population, assessing fitness, establishing breeding elite pools, mutation and crossover, and adjusting for population size diversity. We then use NSGA-II truncation selection and breeding elite pool updates to optimize the symbolic regression model.
While ensuring fitting accuracy, it reduces model complexity, computational and storage costs, and improves model interpretability and search efficiency.
Smart Images
Figure CN118468922B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of intelligent computing and high-performance computing technology, and relates to a symbolic regression method based on genetic programming of elite pools and population size diversity. Background Technology
[0002] Over the past decade, Explainable Artificial Intelligence (XAI) has garnered significant interest due to its importance in key applications such as autonomous vehicles, law, and healthcare. First, the goal of XAI is to ensure that intelligent models provide correct and explainable decisions, enabling human users to understand and trust the AI. This is because AI can exploit "tricks" and misuse coincidences in its training data. A prime example is an image classifier that checks for the presence of wolves in the background of an image, given that the training data consists of numerous images of wolves in the snow. Second, researching explainable AI helps avoid unfairness, bias, and opacity that AI technology might introduce. For instance, automated loan approval systems in banks must provide reasons for rejecting loan applications to satisfy applicants, ensuring fairness and transparency. Third, legislative requirements for explainability are increasingly stringent. For example, the EU's General Data Protection Regulation (GDPR) requires the ability to explain machine learning models. Use cases such as credit scoring and medical imaging demonstrate that machine learning models meet GDPR requirements. Finally, explanations can provide new insights, uncovering new knowledge that surpasses what human scientists have discovered.
[0003] Symbolic regression is a data-driven evolutionary algorithm modeling method based on a syntax tree framework. Examples include Genetic Algorithms (GA), Genetic Programming (GP), and Gene Expression Programming. It automatically optimizes the topology and internal composition of the syntax tree through evolutionary computation. Furthermore, symbolic regression modeling does not require pre-defined model assumptions, thus exhibiting lower structural risk and higher degrees of freedom, which helps improve the generalization performance of the regression model. Symbolic regression models also offer advantages such as good interpretability because they do not require kernel function design.
[0004] Generalized regression (GP) remains an excellent and suitable method for solving symbolic regression problems. However, as the complexity of the problem increases, the search space expands significantly, greatly reducing the search efficiency of traditional GP methods and making it difficult to obtain a solution with reasonable accuracy and acceptable complexity within a limited time. Summary of the Invention
[0005] In view of this, the purpose of this invention is to provide a symbolic regression method based on genetic programming of breeding elite pools and population size diversity, which minimizes model complexity while ensuring that the fitting accuracy is not reduced.
[0006] To achieve the above objectives, the present invention provides the following technical solution:
[0007] A symbolic regression method based on genetic programming of breeding elite pools and population size diversity includes the following steps:
[0008] S1: Initialize the population: Randomly initialize the population, with a population size of Np;
[0009] S2: Population Assessment: Calculate the fitness of all individuals in the population;
[0010] S3: Perform non-dominated sorting of the population and calculate crowding;
[0011] S4: Create and initialize the Breeding Elite Pool (BEP) with a pool length of L. p ;
[0012] S5: Use Nt-tournament to select the parent population;
[0013] S6: Elite Pool Mutation: The subtree of this mutation comes from the elite pool. The mutated subtree is randomly grafted onto a random terminal node in the parent individual to obtain an intermediate offspring population.
[0014] S7: Standard Mutation: Perform standard mutation on the intermediate offspring population in step S6, including standard mutation and standard crossover, to obtain the offspring population;
[0015] S8: Assess the fitness of the offspring population;
[0016] S9: Merge the offspring population and the parent population into one population, and adjust the complexity objective based on population size diversity (SD);
[0017] S10: Perform non-dominated sorting and calculate crowding degree for the offspring population and the population after merging the parents;
[0018] S11: Apply NSGA-II truncation selection to the offspring population and the population after merging the parents to obtain the final offspring population;
[0019] S12: Elite Pool Update: Update the elite pool based on the condition of the offspring population, and determine whether the offspring population has reached the termination criterion; if it has, end the process; otherwise, proceed to step S5.
[0020] Furthermore, in step S1, the Np mathematical expressions are encoded, and Np corresponding syntax trees are randomly generated as a population.
[0021] Furthermore, in step S2, fitness is defined using model accuracy and model complexity;
[0022] The model accuracy is measured using MSE (Mean-squared-error) or RMSE, and the dataset is represented as follows:
[0023]
[0024] Where n is the number of samples, x i These are features in the dataset;
[0025] The precision objective in a symbolic regression task is defined as:
[0026]
[0027] Multi-objective algorithms provide cost-saving pressure;
[0028] Complexity target f c Defined as:
[0029] f c (T) = Size(T)
[0030] Fitness in multi-objective genetic programming is defined as:
[0031] Fitness = (f acc ,f c ).
[0032] Furthermore, step S4, which involves establishing and initializing the elite pool, specifically involves: randomly generating L... p The syntax tree is placed into the elite breeding pool.
[0033] Furthermore, step S6 specifically includes the following steps:
[0034] S61: Select an individual from the population that has not been mutated and is to be mutated;
[0035] S62: Randomly select a subtree from the elite breeding pool, and replace a random terminal node in the individual to be mutated with the subtree to obtain a mutated individual;
[0036] S63: Evaluate the individual to be mutated and the mutated individual. If the accuracy of the mutated individual is better than that of the individual to be mutated, increment the success count of attribute mutation in the subtree (NVS) by one; otherwise, increment the failure count of attribute mutation in the subtree (NVF) by one. Initially, both the success count and the failure count are 0.
[0037] S64: Save the mutated subtree and determine whether all individuals in the population have been mutated. If so, continue execution; otherwise, return to step S61.
[0038] S65: Calculate the mutation success rate (SVR) of all subtrees in the elite breeding pool. The calculation method is as follows, and the mutation success rate is assigned as an attribute to the subtrees in each pool.
[0039]
[0040] Furthermore, step S9 specifically includes the following steps:
[0041] The offspring and parent populations are merged into one population, and the complexity objective for all individuals in this population is modified to population size diversity. The specific calculation method for population size diversity SD is as follows:
[0042]
[0043] Where s is the center size, which is the SD parameter for calculating individuals of size s, l is a hyperparameter representing the distance from the center s, N(s) represents the number of individuals of size s, and A is the number of sizes considered, i.e. the number of terms in the numerator.
[0044] Furthermore, step S12 specifically includes the following steps:
[0045] S121: Select the individuals with the best Nc based on accuracy;
[0046] S122: Find all subtrees among the top Nc individuals: For each Nc individual, find all tree slices with terminal properties.
[0047] S123: Save these subtrees into a dictionary and record their occurrence counts;
[0048] S124: Select candidate subtrees based on occurrence frequency: Find the subtrees in the dictionary that occur more than Ne; if the number of subtrees that meet the requirements is greater than Ns, select the individuals that occur more than Ns times as candidates; if the number of subtrees that meet the requirements is less than or equal to Ns, then all subtrees in the dictionary that occur more than Ne times are candidates.
[0049] S125: Replace the candidate with an individual in the elite breeding pool BEP if the number of individuals in the elite breeding pool is less than the pool length L. p Place all candidates into the pool; if the number of individuals in the elite subtree pool reaches the pool length L... p If the candidate is selected, it will replace the individual with the lowest mutation success rate (SVR) in the elite subtree pool.
[0050] The beneficial effects of this invention are as follows:
[0051] (1) Compared with the traditional multi-objective algorithm, which provides a more economical pressure, this invention can achieve higher fitting accuracy and greater model size diversity.
[0052] (2) Compared with some novel multi-objective algorithms, the present invention can achieve higher fitting accuracy and lower model complexity, making the model more interpretable.
[0053] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description
[0054] To make the objectives, technical solutions, and advantages of the present invention clearer, the preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, wherein:
[0055] Figure 1 This is a flowchart illustrating the overall process of the symbolic regression method based on genetic programming of elite pools and population size diversity in this invention.
[0056] Figure 2 Genetic programming syntax tree;
[0057] Figure 3 For the subtree segments that S0 can generate;
[0058] Figure 4 The flowchart shows the symbolic regression method S12 based on genetic programming of breeding elite pools and population size diversity. Detailed Implementation
[0059] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the illustrations provided in the following embodiments are only schematic representations of the basic concept of the present invention. Unless otherwise specified, the following embodiments and features can be combined with each other.
[0060] The accompanying drawings are for illustrative purposes only and are schematic diagrams, not actual pictures. They should not be construed as limiting the invention. To better illustrate the embodiments of the invention, some parts in the drawings may be omitted, enlarged, or reduced, and do not represent the actual product dimensions. It is understandable to those skilled in the art that some well-known structures and their descriptions may be omitted in the drawings.
[0061] In the accompanying drawings of the embodiments of the present invention, the same or similar reference numerals correspond to the same or similar components. In the description of the present invention, it should be understood that if terms such as "upper," "lower," "left," "right," "front," and "rear" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, they are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, the terms used to describe positional relationships in the drawings are only for illustrative purposes and should not be construed as limiting the present invention. For those skilled in the art, the specific meaning of the above terms can be understood according to the specific circumstances.
[0062] Genetic programming (GP) remains an excellent and suitable method for solving symbolic regression problems. However, as problem complexity increases, the search space expands significantly, drastically reducing the search efficiency of traditional GP methods and making it difficult to obtain a solution with reasonable accuracy and acceptable complexity within a limited time. Furthermore, traditional multi-objective genetic programming algorithms are prone to evolutionary degradation, leading to algorithmic inefficiency. This proposal suggests using a breeding elite pool and diverse population size in multi-objective genetic programming, which reduces model complexity and computational and storage costs while maintaining fitting efficiency. It also addresses the algorithmic inefficiency problem inherent in traditional multi-objective genetic programming frameworks.
[0063] like Figure 1 As shown, this invention provides a symbolic regression method based on genetic programming of elite pools and population size diversity, specifically implemented through the following steps:
[0064] S1: Population Initialization: Randomly initialize the population, with a population size of Np. In the proposed method, each mathematical expression is encoded into a corresponding syntax tree. The nodes of the syntax tree consist of elements from the terminal set and the function set. Population initialization involves randomly generating Np syntax trees as a population. For example... Figure 2 As shown, the syntax tree is represented as x+xy.
[0065] S2: Population Evaluation: Calculate the fitness of all individuals in the population. Evaluating the population involves calculating the fitness of all individuals in the population. Fitness is defined using two objectives (model accuracy and model complexity); accuracy is a common fitness metric in evolutionary algorithms, often using Mean-squared Error (MSE) or Relative Mean Squared Error (RMSE) as the standard. The dataset can be represented as... Where n is the number of samples, x i These are features in the dataset. Thus, the accuracy objective (individual fitness) in a symbolic regression task can be defined as follows:
[0066]
[0067] Multi-objective algorithms provide cost-effectiveness. The complexity objective f... c It can be defined as:
[0068] f c (T) = Size(T)
[0069] Fitness in multi-objective genetic programming can be defined as
[0070] Fitness = (f acc ,f c )
[0071] S3: Perform non-dominated sorting of the population and calculate crowding.
[0072] S4: Establish and initialize the elite breeding pool with a pool length of L. p The specific operation is to randomly generate L. p The syntax tree is placed into the elite breeding pool.
[0073] S5: Tournament selection of parent population: Nt-tournament selection can be used, such as 2-tournament or 3-tournament selection to select the parent population.
[0074] S6: Elite Pool Mutation: This mutation differs from the standard mutation in that the subtree originates from the elite pool instead of being randomly generated. This subtree is randomly grafted onto a random terminal node in the parent individual to obtain an intermediate offspring population. Step S6 is performed as follows:
[0075] S6.1: Select an individual from the population that has not been mutated and is to be mutated.
[0076] S6.2: Randomly select a subtree from the elite breeding pool, and use this subtree to replace a random terminal node in the individual to be mutated, to obtain a mutated individual.
[0077] S6.3: Evaluate the individual to be mutated and the mutated individual. If the accuracy of the mutated individual is better than that of the individual to be mutated, increment the attribute mutation success count (NVS) of the subtree by one; otherwise, increment the attribute mutation failure count (NVF) of the subtree by one. Initially, both the mutation success count and the mutation failure count are 0.
[0078] S6.4: Save the mutated subtree and determine whether all individuals in the population have been mutated. If so, continue execution; otherwise, return to S6.1.
[0079] S6.5: Calculate the mutation success rate (SVR) of all subtrees in the elite breeding pool. The calculation method is as follows, and the mutation success rate is assigned as an attribute to the subtrees in each pool.
[0080]
[0081] S7: Standard Mutation: Perform standard mutation on the intermediate offspring population in S6, including standard mutation and standard crossover (e.g., single-point crossover), to obtain the offspring population.
[0082] S8: Assess the fitness of the offspring population.
[0083] S9: Merge the offspring and parent populations into a single population, adjusting the complexity objective based on population size diversity. Step S9 is executed as follows:
[0084] The offspring and parent populations are merged into a single population, and the complexity objective for all individuals in this population is modified to population size diversity. The specific calculation method for population size diversity (SD) is as follows:
[0085]
[0086] Where s is the center size, i.e., the SD parameter for calculating individuals of size s. l is a hyperparameter representing the distance from the center s, and N(s) represents the number of individuals of size s. A is the number of sizes considered, i.e., the number of terms in the numerator, usually A = 2l + 1.
[0087] For example, l = 1, s = 25:
[0088]
[0089] For example, l = 2, s = 25:
[0090]
[0091] S10: Perform non-dominated sorting and calculate crowding degree for the offspring population and the population after merging the parents.
[0092] S11: Apply NSGA-II truncation selection to the offspring population and the population after merging the parents to obtain the final offspring population.
[0093] S12: Elite Pool Update: Update the elite pool based on the offspring population's condition and determine if the offspring population has met the termination criteria. If it has, the process ends. Otherwise, proceed to step S5. The specific operation of step S12 is as follows: Figure 4 As shown.
[0094] S12.1: The algorithm first selects the top Nc individuals based on accuracy.
[0095] S12.2: This will find all subtrees within these individuals. For example... Figure 3 The selected individual S0 can be used to find all terminal-related slices, such as S1, S2, S3, S4, and S5. Furthermore, based on the required size constraint of L to U, for example, 3 to 80, S1 and S2 can be selected as the desired subtrees.
[0096] S12.3: Save these subtrees into a dictionary and record their occurrence counts. In this example, only S1 and S2 are saved because S3, S4, and S5 do not meet the size limit.
[0097] S12.4: Select candidate subtrees based on frequency of occurrence. Specifically, find the subtrees in the dictionary whose frequency is greater than Ne. If the number of subtrees meeting the requirements is greater than Ns, select the individuals with the highest frequency (Ns) as candidates; if the number of subtrees meeting the requirements is less than or equal to Ns, then all subtrees in the dictionary whose frequency is greater than Ne are candidates. Assume the number of candidates is N. In the experiment, Ne = 3, Ns = 10.
[0098] S12.5: Replace individuals in the Breeding Elite Pool (BEP) with N candidates. If the number of individuals in the Breeding Elite Pool is less than the pool length (L) defined in S4. p Then, simply place all candidates into the pool. If the number of individuals in the elite subtree pool reaches the pool length (L)... p If a candidate is selected, it will replace the individual with the lowest success rate (SVR) in the elite subtree breeding pool. This method ensures that the number of subtrees in the pool does not exceed the pool length (L). p In the experiment, the pool length (L) p The value is 50.
[0099] Another embodiment of the present invention provides a detailed description of the application conditions and effects of the present invention in conjunction with simulation.
[0100] (1) Simulation conditions
[0101] The main objective of this work is to reduce the complexity of the symbolic regression model while maintaining fitting accuracy. We need to pre-set the basic parameters of the genetic programming. These parameters include population size, tournament selection size, generation, crossover-mutation proportion, initialization method, maximum individual size, function set, and terminal set.
[0102] In addition, this work requires setting some hyperparameters for breeding elite pools and population size diversity, including L, U, and L. p ,Ne,Nc,Ns,l。
[0103] The specific experimental setup for the simulation is shown in Table 1.
[0104] Table 1
[0105]
[0106] The datasets used for testing symbolic regression include Airfoil, Boston, Concrete, Dow Chemical, Energy:cooling, Energy:heating, and Yacht. All datasets were randomly split into a 4:1 ratio for training and testing. Furthermore, all datasets were normalized to a mean of 0 and a variance of 1.
[0107] In contrast, traditional multi-objective genetic programming algorithms, as well as recently proposed algorithms, will be considered. The comparison method is as follows.
[0108] The traditional multi-objective genetic programming algorithms NSGA-II and SPEAII will be tested to evaluate the accuracy and complexity performance of the output models. Due to the use of a second objective to provide cost-saving pressure, some data suggests that this can lead to evolutionary inefficiency, making it difficult to find highly accurate individuals, resulting in premature convergence and getting trapped in local optima.
[0109] Traditional single-objective genetic programming will also be used as a comparison algorithm. Due to the lack of constraints on complexity, this algorithm often leads to a severe bloat problem. This results in a significant waste of computer computing and storage resources.
[0110] In addition, a recently proposed algorithm, Adaptive-alpha Dominance, will be used as a comparison algorithm. This algorithm adjusts the selection pressure between large and small individuals based on the population situation, preventing it from getting trapped in local optima like traditional algorithms such as NSGA-II. It also alleviates the bloat problem of traditional single-objective genetic programming.
[0111] This experiment will run the comparison algorithm and the algorithm proposed in this work 30 times each on the above 7 datasets to reduce the error.
[0112] (2) Simulation results
[0113] Table 2
[0114]
[0115] Table 3
[0116]
[0117] Table 2 shows the MSE (accuracy) performance of various algorithms on the test set, and Table 3 shows the performance of various algorithms on the optimal individual size (complexity) on the test set. Tables 2 and 3 describe the accuracy and complexity performance of each algorithm. Among them, the symbolic regression method based on breeding an elite pool and population size diversity genetic programming (NSGA-II with BEP+SD in the table) performs best in accuracy, followed by adaptive alpha dominance. NSGA-II and SPEA-II perform very poorly, due to inefficiencies or population collapse problems described in some literature.
[0118] In terms of complexity, the symbolic regression method based on breeding elite pools and population size diversity genetic programming produces models with acceptable complexity, better than adaptive alpha dominance and standard genetic programming. While NSGA-II and SPEA-II produce models with very low complexity, their model accuracy is also very low.
[0119] Therefore, it can be concluded that the symbolic regression method based on breeding elite pools and population size diversity genetic programming can reduce model complexity while ensuring fitting accuracy.
[0120] Those skilled in the art will understand that all or part of the steps in the methods of the above embodiments can be implemented by a program instructing related hardware. The program can be stored in a computer-readable storage medium. When the program is executed, it can implement the steps of the method. The storage medium may be, for example, ROM / RAM, magnetic disk, optical disk, etc.
[0121] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.
Claims
1. A symbolic regression method based on genetic programming of elite pools and population size diversity, characterized in that: Includes the following steps: S1: Initialize the population: Randomly initialize the population with a size of Np; S2: Population Assessment: Calculate the fitness of all individuals in the population; S3: Perform non-dominated sorting of the population and calculate crowding; S4: Establish and initialize the elite training pool, with a pool length of [missing information]. ; S5: Use Nt-tournament to select the parent population; S6: Elite Pool Mutation: The mutated subtree comes from the elite pool. This mutated subtree is randomly grafted onto a random terminal node in the parent individual to obtain an intermediate offspring population. Step S6 specifically includes the following steps: S61: Select an individual from the population that has not been mutated and is to be mutated; S62: Randomly select a subtree from the elite breeding pool, and replace a random terminal node in the individual to be mutated with the subtree to obtain a mutated individual; S63: Evaluate the individual to be mutated and the mutated individual. If the accuracy of the mutated individual is better than that of the individual to be mutated, increment the success count of attribute mutation in the subtree (NVS) by one; otherwise, increment the failure count of attribute mutation in the subtree (NVF) by one. Initially, both the success count and the failure count are 0. S64: Save the mutated subtree and determine whether all individuals in the population have been mutated. If so, continue execution; otherwise, return to step S61. S65: Calculate the mutation success rate (SVR) of all subtrees in the elite breeding pool. The calculation method is as follows, and the mutation success rate is assigned as an attribute to the subtrees in each pool. S7: Standard Mutation: Perform standard mutation on the intermediate offspring population in step S6, including standard mutation and standard crossover, to obtain the offspring population; S8: Assess the fitness of the offspring population; S9: Merge the offspring and parent populations into a single population, adjusting the complexity objective based on population size diversity; Step S9 specifically includes the following steps: The offspring and parent populations are merged into one population, and the complexity objective for all individuals in this population is modified to population size diversity. The specific calculation method for population size diversity SD is as follows: Where s is the center size, i.e., the SD parameter for calculating an individual of size s. It is a hyperparameter representing the distance from the center s. The number of individuals of size s is represented by A, which is the number of sizes considered, i.e., the number of terms in the numerator. S10: Perform non-dominated sorting and calculate crowding degree on the offspring population and the population after merging the parents; S11: Apply NSGA-II truncation selection to the offspring population and the population after merging the parents to obtain the final offspring population; S12: Elite Pool Update: Update the elite pool based on the offspring population and determine whether the offspring population has reached the termination criterion; if it has, end the process; otherwise, proceed to step S5. Step S12 specifically includes the following steps: S121: Select the individuals with the best Nc based on accuracy; S122: Find all subtrees among the top Nc individuals: For each Nc individual, find all tree slices with terminal properties. S123: Save these subtrees into a dictionary and record their occurrence counts; S124: Select candidate subtrees based on occurrence frequency: Find the subtrees in the dictionary that occur more than Ne; if the number of subtrees that meet the requirements is greater than Ns, select the individuals that occur more than Ns times as candidates; if the number of subtrees that meet the requirements is less than or equal to Ns, then all subtrees in the dictionary that occur more than Ne times are candidates. S125: Replace the candidate with an individual in the Elite Pool (BEP) if the number of individuals in the Elite Pool is less than the pool length. Place all candidates into the pool; if the number of individuals in the elite subtree pool reaches the pool length... If the candidate is selected, it will replace the individual with the lowest mutation success rate (SVR) in the elite subtree pool.
2. The symbolic regression method based on genetic programming of elite pools and population size diversity according to claim 1, characterized in that: In step S1, the Np mathematical expressions are encoded, and Np corresponding syntax trees are randomly generated as a population.
3. The symbolic regression method based on genetic programming of elite pools and population size diversity according to claim 1, characterized in that: In step S2, fitness is defined using model accuracy and model complexity; The model accuracy is measured using MSE or RMSE, and the dataset is represented as follows: in For the sample size, These are features in the dataset; The precision objective in a symbolic regression task is defined as: Multi-objective algorithms provide cost-saving pressure; Complexity target Defined as: Fitness in multi-objective genetic programming is defined as:
4. The symbolic regression method based on genetic programming of elite pools and population size diversity according to claim 1, characterized in that: Step S4, establishing and initializing the elite pool, specifically involves: randomly generating... The syntax tree is placed into the elite breeding pool.