Bridge scour formula optimized reconstruction method based on symbol regression

Abstract

A bridge scour formula optimized reconstruction method based on symbol regression. The method comprises: obtaining statistical data of bridge scour, and separately using the statistical data as a training set and a test set; determining an initial structure of a bridge scour formula; firstly, determining a candidate operator, then introducing a function group having significant physical importance and a statistical relationship in scour, and determining a candidate operation variable; by using a symbol regression method based on genetic programming, generating an optimization formula by using data of the training set; adjusting parameters and generating an alternative formula group; and, on the basis of the test set, performing formula effect evaluation and filtering out a formula form having an optimal predicted effect and the simplest formula structure. On the basis of existing standard formulas, by using a symbol regression method based on a genetic programming algorithm, a machine learning method having a nonlinear representation capability is fused, in order to more accurately mine rich features in data. Meanwhile, by combining prior knowledge of scour and a physical relationship, prediction performance and generalization of a scour calculation formula can be improved.

Description

A bridge scour formula optimization and reconstruction method based on symbolic regression Technical Field

[0001] The invention belongs to the technical field of civil engineering and water conservancy projects, and in particular relates to a bridge scour formula optimization reconstruction method based on symbolic regression. Background Art

[0002] Local scour of bridge piers is a phenomenon in which water erodes the surrounding sediment due to the obstruction of the piers. It involves a complex interaction between water flow, sediment, and the structure. According to relevant literature, approximately 50% of bridge failures in recent years have been caused by scour of the bridge foundation structure and its associated hydraulic effects, resulting in significant economic losses. Therefore, estimating the local scour depth of bridges is a key component of bridge safety assessment. Currently, scholars both domestically and internationally have proposed numerous formulas for calculating the local scour depth based on the various factors affecting local scour of bridge piers. Most of these are empirical or semi-empirical formulas based on field observations and test data. These results are often only accurate in specific circumstances and lack stability and generalizability. Some formulas produce overly conservative results, which are detrimental to the economic efficiency of engineering designs; others produce underestimating results, which can pose serious safety risks to bridge scour designs.

[0003] Summary of the Invention

[0004] Purpose of the invention: In response to the problems of accuracy and universality of existing pier scour formulas, the purpose of the present invention is to provide a bridge scour formula optimization and reconstruction method based on symbolic regression. Based on the existing standard formula, this method combines prior knowledge and physical relationships of pier scour, uses a symbolic regression method based on a genetic programming algorithm, learns from a large amount of measured data, discovers the complex relationship between scour depth and various influencing factors, and obtains a formula with good predictive performance and is easy to understand and use.

[0005] Technical solution: To achieve the above objectives, the present invention discloses a bridge scour formula optimization and reconstruction method based on symbolic regression, comprising the following steps:

[0006] (1) Obtain statistical data on bridge scour, which are used as training and test sets respectively;

[0007] (2) Determine the initial structure of the bridge scour formula;

[0008] (3) First, determine the candidate operators, then introduce a group of functions with significant physical meanings and statistical relationships in the scour to determine the candidate operating variables;

[0009] (4) Using a symbolic regression method based on genetic programming, the optimization formula is generated using the data of the training set;

[0010] (5) Adjust parameters and generate a group of alternative formulas

[0011] (6) Evaluate the formula effect based on the test set and select the formula with the best prediction effect and the simplest formula structure.

[0012] In step (1), statistical data are obtained from the literature of previous years as training set and test set respectively, and it is ensured that the references of training set and test set do not overlap and the data are independent of each other; the statistical data include: Data{b,L,θ,V1,y1,d 50 ,σ g ,y s}

[0013] b is the width of the pier, L is the length of the bridge, θ is the angle of attack of the water flow, V1 is the average velocity of the water flow upstream of the pier, y1 is the water depth upstream of the pier after general scouring, d 50 is the sediment particle size, σ g is the sediment uniformity, y s It is the local scouring depth of the bridge pier.

[0014] Preferably, in step (2), a new calculation formula is generated by optimizing and reconstructing the CSU equation structure adopted in the US standard:

[0015] Where: y s is the local scouring depth of the pier, y1 is the water depth upstream of the pier after general scouring, K1 is the pier shape correction coefficient, K2 is the water flow angle correction coefficient, K3 is the riverbed condition correction coefficient, b is the pier width, F r is the Froude number of the water flow upstream of the pier, Where V1 is the average velocity of water flow upstream of the pier, and g is the acceleration due to gravity;

[0016] Pair both sides of the formula:

[0017] The initial structure of the optimized bridge scour formula is as follows:

[0018] Furthermore, in step (3), common operator symbols are used to determine the candidate operators including: Operator{+,-,×,÷,^2,^3,sin,cos}

[0019] Combining the bridge scour test results with correlation analysis, it was determined that the following parameters have a significant statistical relationship with the scour depth: Parameter {b n ,d 50 ,σ g ,y1,V1}

[0020] Among them, b nis the effective width, b n =bcosθ+Lsinθ, b is the pier width, l is the bridge length, θ is the water flow angle, d 50 is the sediment particle size, σ g is the sediment uniformity, y1 is the water depth upstream of the pier after general scouring, and V1 is the average water velocity upstream of the pier;

[0021] Local scour depth of bridge pier y s The relationship with Parameter is as follows:

[0022] According to the analysis of the scour test results, the above function group can be used: g1(x,y)=x+y, g2(x)=sin(ax), g3(x)=b x (0<b<1), g4(x)=x c-x , g5 = log d (x+1)(d>1); parameters a, b, c, d should be determined according to the data set;

[0023] Candidate manipulation variables include:

[0024] Furthermore, step (4) specifically includes the following steps:

[0025] (4.1) Based on the candidate operators and candidate operation variables obtained in step (3), a random initial expression population containing operators, constants and variables is generated. The population size i is defined as the number of individuals in the population. The initialization process needs to be repeated until i expressions are generated: Original{E1,E2,…,E i};

[0026] (4.2) Based on the training set data, the fitness function is used to evaluate the population performance, and the top n expressions with the best performance are selected as the parent population of the next generation, Parents{E1, E2, …, E n}, n = 0.8i;

[0027] (4.3) Based on the training set data, the genetic programming algorithm is used to perform genetic iteration on the selected population to generate the sub-population with the best fitness. The process is divided into three stages: crossover stage, mutation stage, and replacement stage.

[0028] (4.4) Calculate the new population New{E1,E2,…,E n} fitness, and the new population New{E1,E2,…,E n} Sort by fitness from small to large and use them as the parent population for the next iteration Parents{E1,E2,…,E n}; Return to step (4.3), and after reaching the set number of iterations, take out the expression with the best fitness in the population as the target formula E target .

[0029] Preferably, the fitness calculation formula for each expression in step (4.2) is as follows:

[0030] Where N is the total number of samples, It is individual E i The predicted value of the jth sample, y i is the corresponding true value.

[0031] Furthermore, in step (4.3), the crossover phase refers to randomly selecting subtrees of two parent populations and exchanging them to generate two new offspring trees; the mutation phase refers to randomly mutating the nodes of the individual expression trees while keeping the tree structure unchanged; the replacement phase refers to regenerating a new tree to replace the current tree.

[0032] Furthermore, the target formula E in step (4.4) target for:

[0033] Preferably, in step 5, the parameter set P can be adjusted: P = {num, depth, p variation ,p crossover ,p mutation ,preplacement,run}

[0034] Among them, num is the number of initialization expressions, depth is the depth of the tree, p variation is the probability that a node is a variable when generating a tree, p crossover is the probability of crossover between different expressions, p mutation is the probability of node mutation, preplacement is the probability of regenerating a tree to replace the current tree, and run is the number of genetic iterations.

[0035] Furthermore, in step (6), by calculating the candidate formula group MSE and R 2 Sort the prediction performance:

[0036] Where N is the total number of samples, It is individual E i The predicted value of the jth sample, y i is the corresponding true value, is the average value of the true value; the smaller the MSE, the better the prediction effect; R 2 The closer it is to 1, the better the prediction effect.

[0037] Beneficial effects: Compared with the existing technology, the present invention has the following significant advantages: Based on the existing standard formula, the present invention combines the prior knowledge and physical relationship of bridge scour, uses the symbolic regression method based on the genetic programming algorithm, learns from a large amount of measured data, discovers the complex relationship between scour depth and various influencing factors, and obtains a formula with good predictive performance and is easy to understand and use; the present invention overcomes the shortcomings of traditional statistical methods that are difficult to accurately mine the rich features in the data and the limited amount of data, and the optimization formula is more accurate and generalizable than the traditional formula. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] FIG1 is a flow chart of the present invention;

[0039] FIG2 is a flow chart of a symbolic regression method based on a genetic algorithm in the present invention;

[0040] Figure 3 shows the scouring depth y in the present invention. s and sediment particle size d 50 Schematic diagram of the change relationship;

[0041] Figure 4 shows the scouring depth y in the present invention. s and sediment uniformity σ g Schematic diagram of the change relationship;

[0042] Figure 5 shows the scouring depth y in the present invention. s Schematic diagram of the relationship with the change of water flow depth y1;

[0043] Figure 6 shows the scouring depth y in the present invention. s Schematic diagram of the relationship with the change of water flow velocity V1. DETAILED DESCRIPTION

[0044] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0045] Symbolic regression is a supervised learning method that attempts to discover a hidden mathematical formula to predict a target variable using feature variables. Genetic programming is the predominant algorithm for solving symbolic regression problems. Genetic programming is a type of intelligent computing optimization method that simulates the natural evolutionary process of "survival of the fittest" to enable autonomous computer programming. Genetic programming algorithms offer a flexible encoding mechanism, making them suitable for representing the structure of evolving functions. Furthermore, the algorithm's search process is based on random heuristic search, eliminating the need for complex optimization models and making it highly versatile.

[0046] By optimizing and reconstructing existing scour formulas using symbolic regression, and building upon existing empirical formulas and incorporating prior knowledge of scour and physical relationships, a more accurate and universal scour depth prediction model can be constructed. This scour depth prediction model can better adapt to varying river conditions, pier shapes, and fluid dynamics. The resulting formula not only improves predictive performance but is also easier for engineers to understand and use, thereby enhancing the safety and cost-effectiveness of bridge design.

[0047] In summary, applying symbolic regression to the study of local scour depths on bridge piers not only addresses the shortcomings of existing empirical formulas but also provides strong technical support for bridge scour protection design and maintenance. This approach helps reduce bridge damage accidents caused by scour, ensures traffic safety, and reduces maintenance costs. It has significant practical significance and broad application prospects in the field of bridge engineering.

[0048] As shown in FIG1 , the bridge scour formula optimization and reconstruction method based on symbolic regression of the present invention includes the following steps:

[0049] (1) Obtain statistical data on bridge scour, which are used as training and test sets respectively;

[0050] In step (1), statistical data are obtained from the literature of previous years as training set and test set respectively, and it is ensured that the references of training set and test set do not overlap and the data are independent of each other; the statistical data include: Data{b,L,θ,V1,y1,d 50 ,σ g ,y s}

[0051] b is the width of the pier, L is the length of the bridge, θ is the angle of attack of the water flow, V1 is the average velocity of the water flow upstream of the pier, y1 is the water depth upstream of the pier after general scouring, d 50 is the sediment particle size, σ g is the sediment uniformity, y s is the local scour depth of the pier;

[0052] (2) Determine the initial structure of the bridge scour formula;

[0053] In step (2), the CSU equation structure adopted by the US standard is optimized and reconstructed to generate a new calculation formula:

[0054] Where: y s is the local scouring depth of the pier, y1 is the water depth upstream of the pier after general scouring, K1 is the pier shape correction coefficient, K2 is the water flow angle correction coefficient, K3 is the riverbed condition correction coefficient, b is the pier width, F r is the Froude number of the water flow upstream of the pier, Where V1 is the average velocity of water flow upstream of the pier, and g is the acceleration due to gravity;

[0055] Pair both sides of the formula:

[0056] The initial structure of the optimized bridge scour formula is as follows:

[0057] (3) First, determine the candidate operators, then introduce a group of functions with significant physical meanings and statistical relationships in the scour to determine the candidate operating variables;

[0058] As shown in FIG2 , common operator symbols are used in step (3), and the candidate operators determined include: Operator{+,-,×,÷,^2,^3,sin,cos}

[0059] Combining the bridge scour test results with correlation analysis, it was determined that the following parameters have a significant statistical relationship with the scour depth: Parameter {b n ,d 50 ,σ g ,y1,V1}

[0060] Among them, b n is the effective width, b n =bcosθ+Lsinθ, b is the width of the pier, L is the length of the bridge, θ is the angle of attack of the water flow, d 50 is the sediment particle size, σ g is the sediment uniformity, y1 is the water depth upstream of the pier after general scouring, and V1 is the average water velocity upstream of the pier;

[0061] As shown in Figures 3, 4, 5 and 6, the local scour depth y of the pier s The relationship with Parameter is as follows:

[0062] According to the analysis of the scour test results, the above function group can be used: g1(x,y)=x+y, g2(x)=sin(ax), g3(x)=b x (0<b<1), g4(x)=x c-x , g5 = log d (x+1)(d>1); parameters a, b, c, d should be determined according to the data set;

[0063] The present invention suggests that candidate operating variables include:

[0064] (4) Using the symbolic regression method based on genetic programming, the optimization formula is generated using the data of the training set. The specific steps are as follows:

[0065] (4.1) Based on the candidate operators and candidate operation variables obtained in step (3), a random initial expression population containing operators, constants and variables is generated. The population size i is defined as the number of individuals in the population. The initialization process needs to be repeated until i expressions are generated: original {E1, E2, …, E i};

[0066] (4.2) Based on the training set data, the fitness function is used to evaluate the population performance, and the top n expressions with the best performance are selected as the parent population of the next generation, Parents{E1, E2, …, E n}, n = 0.8i;

[0067] The fitness calculation formula for each expression is as follows:

[0068] Where N is the total number of samples, It is individual E i The predicted value of the jth sample, y i is the corresponding true value;

[0069] (4.3) Based on the training set data, the genetic programming algorithm is used to perform genetic iteration on the selected population to generate the sub-population with the best fitness. It is divided into three stages:

[0070] Crossover: Randomly select subtrees of two maternal populations and exchange them to generate two new offspring trees;

[0071] Mutation: Randomly mutate the nodes of individual expression trees while keeping the tree structure unchanged;

[0072] Replace: Regenerate a new tree to replace the current tree;

[0073] (4.4) Calculate the new population New{E1,E2,…,E n} fitness, and the new population New{E1,E2,…,E n} Sort by fitness from small to large and use them as the parent population for the next iteration parents{E1,E2,…,E n}; Return to step (4.3), and after reaching the set number of iterations, take out the expression with the best fitness in the population as the target formula E target ;

[0074] Target Formula E target as follows:

[0075] (5) Adjust parameters and generate a group of alternative formulas

[0076] In step 5, the parameter set P can be adjusted: P = {num, depth, p variation ,p crossover ,p mutation ,preplacement,run}

[0077] Among them, num is the number of initialization expressions, depth is the depth of the tree, p variation is the probability that a node is a variable when generating a tree, p crossover is the probability of crossover between different expressions, p mutation is the probability of node mutation, preplacement is the probability of regenerating a tree to replace the current tree, and run is the number of genetic iterations;

[0078] (6) Evaluate the formula effect based on the test set and select the formula with the best prediction effect and the simplest formula structure;

[0079] In step (6), the candidate formula group is calculated MSE and R 2 Sort the prediction performance:

[0080] Where N is the total number of samples, It is individual E i The predicted value of the jth sample, y i is the corresponding true value, is the average value of the true value; the smaller the MSE, the better the prediction effect; R 2 The closer it is to 1, the better the prediction effect.

Claims

1. A bridge scour formula optimization and reconstruction method based on symbolic regression, characterized by: The steps include: (1) Obtain statistical data on bridge scour, which are used as training and test sets respectively; (2) Determine the initial structure of the bridge scour formula; (3) First, determine the candidate operators, then introduce a group of functions with significant physical meanings and statistical relationships in the scour to determine the candidate operating variables; (4) Using a symbolic regression method based on genetic programming, the optimization formula is generated using the data of the training set; (5) Adjust parameters and generate a group of alternative formulas (6) Evaluate the formula effect based on the test set and select the formula with the best prediction effect and the simplest formula structure.

2. The bridge scour formula optimization and reconstruction method based on symbolic regression according to claim 1 is characterized by: In step (1), statistical data are obtained from the literature of previous years as training set and test set respectively, and it is ensured that the references of training set and test set do not overlap and the data are independent of each other; the statistical data include: Data{b,L,θ,V1,y1,d 50 ,σ g ,y s } b is the width of the pier, L is the length of the bridge, θ is the angle of attack of the water flow, V1 is the average velocity of the water flow upstream of the pier, y1 is the water depth upstream of the pier after general scouring, d 50 is the sediment particle size, σ g is the sediment uniformity, y s It is the local scouring depth of the bridge pier.

3. The bridge scour formula optimization and reconstruction method based on symbolic regression according to claim 2 is characterized by: In step (2), the CSU equation structure adopted by the US standard is optimized and reconstructed to generate a new calculation formula: Where: y s is the local scouring depth of the pier, y1 is the water depth upstream of the pier after general scouring, K1 is the pier shape correction coefficient, K2 is the water flow angle correction coefficient, K3 is the riverbed condition correction coefficient, b is the pier width, F r is the Froude number of the water flow upstream of the pier, Where B1 is the average velocity of water flow upstream of the pier, and g is the acceleration due to gravity; Pair both sides of the formula: The initial structure of the optimized bridge scour formula is as follows:

4. The bridge scour formula optimization and reconstruction method based on symbolic regression according to claim 3 is characterized by: In step (3), common operator symbols are used to determine the candidate operators including: Operator{+,-,×,÷,^2,^3,sin,cos} Combining the bridge scour test results with correlation analysis, it was determined that the following parameters have a significant statistical relationship with the scour depth: Parameter{b n ,d 50 ,s g ,y1,V1} Among them, b n is the effective width, b n =bcosθ+Lsinθ, b is the width of the pier, L is the length of the bridge, θ is the angle of attack of the water flow, d 50 is the sediment particle size, σ g is the sediment uniformity, y1 is the water depth upstream of the pier after general scouring, and V1 is the average water velocity upstream of the pier; Local scour depth of bridge pier y s The relationship with Parameter is as follows: According to the analysis of the scour test results, the above function group can be used: g1(x,y)=x+y, g2(x)=sin(ax), g3(x)=b x (0 <b<1),g4(x)=x c-x , g5 = log d (x+1)(d>1); parameters a, b, c, d should be determined according to the data set; Candidate manipulation variables include:

5. The bridge scour formula optimization and reconstruction method based on symbolic regression according to claim 4 is characterized by: The step (4) specifically includes the following steps: (4.1) Based on the candidate operators and candidate operation variables obtained in step (3), a random initial expression population containing operators, constants and variables is generated. The population size i is defined as the number of individuals in the population. The initialization process needs to be repeated until i expressions are generated: Original{E1,E2,…,E i }; (4.2) Based on the training set data, the fitness function is used to evaluate the population performance, and the top n expressions with the best performance are selected as the parent population of the next generation, Parents{E1, E2, …, E n }, n = 0.8i; (4.3) Based on the training set data, the genetic programming algorithm is used to perform genetic iteration on the selected population to generate the adaptive The subpopulation with the best degree is divided into three stages, namely the crossover stage, the mutation stage and the replacement stage; (4.4) Calculate the new population New{E1,E2,…,E n } fitness, and the new population New{E1,E2,…,E n } Sort by fitness from small to large and use them as the parent population for the next iteration Parents{E1,E2,…,E n }; Return to step (4.3), and after reaching the set number of iterations, take out the expression with the best fitness in the population as the target formula E target .

6. The bridge scour formula optimization and reconstruction method based on symbolic regression according to claim 5 is characterized by: The fitness calculation formula for each expression in step (4.2) is as follows: Where N is the total number of samples, It is individual E i The predicted value of the jth sample, y i is the corresponding true value.

7. The bridge scour formula optimization and reconstruction method based on symbolic regression according to claim 6 is characterized by: In the step (4.3), the crossover phase refers to randomly selecting subtrees of two parent populations and exchanging them to generate two new offspring trees; the mutation phase refers to randomly mutating the nodes of the individual expression trees while keeping the tree structure unchanged; and the replacement phase refers to regenerating a new tree to replace the current tree.

8. The bridge scour formula optimization and reconstruction method based on symbolic regression according to claim 7 is characterized by: The target formula E in step (4.4) target for:

9. The bridge scour formula optimization and reconstruction method based on symbolic regression according to claim 8 is characterized by: The adjustable parameter set P in step 5 is: P = {num, depth, p variation ,p crossover ,p mutation ,preplacement,run} Among them, num is the number of initialization expressions, depth is the depth of the tree, p variation is the probability that a node is a variable when generating a tree, p crossover is the probability of crossover between different expressions, p mutation is the probability of node mutation, preplacement is the probability of regenerating a tree to replace the current tree, and run is the number of genetic iterations.

10. The bridge scour formula optimization and reconstruction method based on symbolic regression according to claim 9 is characterized by: In step (6), the candidate formula group is calculated MSE and R 2 Sort the prediction performance: Where N is the total number of samples, It is individual E i The predicted value of the jth sample, y i is the corresponding true value, is the average value of the true value; the smaller the MSE, the better the prediction effect; R 2 The closer it is to 1, the better the prediction effect.

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