An intelligent optimization method for the hanger force during the construction stage of a suspension bridge based on gradient boosting

Abstract

An intelligent optimization method for the hanger force during the construction stage of a suspension bridge based on gradient boosting, which relates to the field of intelligent optimization technology for bridge construction. The steps of this method are as follows: data preparation; model definition and optimization objective; training the first tree; determining the splitting point based on gain; updating the leaf nodes; iterative training; predicting new data; error analysis; optimizing the hanger force during the construction stage using the particle swarm optimization algorithm; and outputting the optimization result. It can efficiently establish an accurate mapping relationship between the hanger force during the construction stage and the hanger force at the completed bridge stage, and quickly reverse-calculate the optimal hanger force distribution during the construction stage through a global optimization algorithm, comprehensively improving the optimization efficiency and accuracy.

Description

Technical Field

[0001] The present invention relates to the technical field of intelligent optimization of bridge construction, and specifically to an intelligent optimization method for the hanger force in the construction stage of a suspension bridge based on gradient boosting. Background Art

[0002] As an important structural form in modern bridge engineering, suspension bridges are widely used in various complex terrains and long-span bridge projects due to their large span and high performance. The hanger is an important load-bearing component connecting the main cable and the bridge deck system in a suspension bridge. The hanger force in the construction stage not only affects the bridge alignment of the completed bridge but also relates to the overall safety, durability, and operation performance of the bridge. Therefore, how to reasonably distribute the hanger force in the construction stage to ensure that the hanger force in the completed bridge stage meets the design objectives is a key technical issue in the design and construction of suspension bridges.

[0003] The traditional optimization of the hanger force in the construction stage usually adopts a method combining finite element analysis, manual trial calculation, and repeated iteration. By establishing a finite element model of the suspension bridge and performing iterative calculations, the hanger force in the completed bridge stage is calculated, and the hanger force in the construction stage is inversely deduced therefrom. However, finite element analysis needs to process large-scale bridge models, requires high computer performance, and takes a long time, making it difficult to meet the need for rapid adjustment during the construction process. Moreover, due to model simplification and parameter uncertainty, the results of finite element analysis may deviate from the actual situation, resulting in unsatisfactory optimization effects. Therefore, the current optimization adjustment of the hanger force mainly relies on construction experience for trial-and-error repeated adjustments, lacking systematicness and automation, and is prone to introducing human errors. In addition, there is a complex non-linear mapping relationship between the hanger force in the construction stage and the completed bridge stage. Traditional optimization methods are difficult to accurately describe this relationship, and with the continuous increase in the bridge span and the rapid development of construction technology, the optimization problem of the hanger force gradually shows multi-dimensional and high-complexity characteristics. Traditional optimization methods are prone to falling into local optima in a high-dimensional search space and lack global search capabilities, resulting in low optimization efficiency and difficult to guarantee the quality of the optimization results.

[0004] In summary, traditional optimization methods have defects in the optimization of the hanger force in the construction stage, such as long calculation time, insufficient accuracy, relying on manual adjustment, and difficulty in dealing with non-linear problems. Therefore, there is an urgent need for a new intelligent optimization method to effectively solve the deficiencies of traditional optimization methods. Summary of the Invention

[0005] To solve the deficiencies in the background art, the present invention provides an intelligent optimization method for the hanger force in the construction stage of a suspension bridge based on gradient boosting, which can efficiently establish an accurate mapping relationship between the hanger force in the construction stage and the completed bridge stage, and quickly inversely solve the optimal hanger force distribution in the construction stage through a global optimization algorithm, comprehensively improving the optimization efficiency and accuracy.

[0006] To achieve the above object, the present invention adopts the following technical solutions: An intelligent optimization method for the hanger force in the construction stage of a suspension bridge based on gradient boosting, comprising the following steps:

[0007] Step 1: Data preparation

[0008] Suppose there are m hangers in the construction stage of the suspension bridge, and the number of samples is n. For the i-th sample, the hanger force in the construction stage is expressed as: X i =[x i1 ,x i2 ,…,x im , where x ij represents the hanger force in the construction stage of the j-th hanger of the i-th sample. The corresponding hanger force in the completed bridge stage is expressed as: Y i =[y i1 ,y i2 ,…,y im , where y ij represents the hanger force in the completed bridge stage of the j-th hanger of the i-th sample, i = 1, 2, …, n, j = 1, 2, …, m. The input dataset of the hanger force in the construction stage is: X = {X1, X2, …, X n}, and the output dataset of the hanger force in the completed bridge stage is: Y = {Y1, Y2, …, Y n};

[0009] Step 2: Model definition and optimization objective

[0010] Model mapping relationship: Establish a mapping function f to map the hanger force in the construction stage to the hanger force in the completed bridge stage. The objective of the model is to learn f to minimize the error between the predicted value and the true value Y i ;

[0011] Optimization objective: Use the squared error as the loss function L, represents the predicted value of y ij ;

[0012] Objective formula: Minimize the loss function which is:

[0013] Step 3: Train the first tree

[0014] Initialize the predicted value: In the initial state, the model predicted value is the mean of the objective function,

[0015] Calculate the initial residual: The residual is expressed as the difference between the true value y ij and the current model predicted value :

[0016] Step 4: Determine the splitting point based on the gain

[0017] Gradient and second derivative: Using the second-order Taylor expansion of the loss function, calculate the gradient and second derivative of each sample, which are expressed as follows:

[0018]

[0019] For the sample set D within the current node, we have:

[0020]

[0021] Gain formula: For the splitting point v t , divide the data into the left child node D L and the right child node D R , we have:

[0022]

[0023] In the formula, G L and G R represent the sum of gradients of the left and right child nodes respectively, H L and H R represent the sum of second derivatives of the left and right child nodes respectively, λ represents the regularization parameter, and γ represents the splitting penalty term;

[0024] Select the best splitting point: Traverse all splitting points v t , and select the splitting point with the maximum gain:

[0025] Step 5: Update the leaf nodes

[0026] Output value of the leaf node: For the leaf node, its output value is:

[0027] Update the predicted value: The model predicted value is updated to: η represents the learning rate;

[0028] Step 6: Iterative training

[0029] Recalculate the residuals:

[0030] Termination condition: Reach the maximum number of trees T or the decrease in the loss function is less than the threshold ε;

[0031] Step 7: Predict new data

[0032] Input new data: X new = [x new,1 , x new,2 , …, x new,m;

[0033] Model prediction: Using the cumulative results of all the trees, calculate the new predicted value:

[0034]

[0035] In the formula, T κ (·) represents the prediction function of the κ-th tree, and T κ (X new ) represents the prediction result made by the κ-th tree on the new input data X new ;

[0036] Step Eight: Error analysis

[0037] Compare the suspender forces at the completed bridge stage predicted by the model and the ideal suspender forces Y at the completed bridge stage * , and calculate the error using the following formula:

[0038]

[0039] In the formula, is the suspender force of the j-th suspender at the completed bridge stage predicted, is the suspender force of the j-th suspender at the completed bridge stage of the ideal target;

[0040] If the error E is greater than the allowable threshold set by the project, further adjust X new so that the suspender forces at the completed bridge stage meet the ideal target;

[0041] Step Nine: Optimize the suspender forces during the construction stage using the particle swarm optimization algorithm

[0042] Optimization objective definition:

[0043] 1) Objective function: Define that the optimized suspender forces X during the construction stage need to minimize the error between the predicted suspender forces at the completed bridge stage and the ideal target. The objective function is as follows:

[0044]

[0045] In the formula, f(X) j represents the predicted suspender force of the j-th suspender at the completed bridge stage after optimization;

[0046] 2) Constraint conditions: Define the upper and lower limit constraints for a single suspender:

[0047]

[0048] In the formula, and respectively represent the minimum allowable value and the maximum allowable value of the j-th suspender force;

[0049] Particle Swarm Optimization Algorithm:

[0050] 1) Particle initialization: Initialize N particles, where each particle represents a set of suspension rod forces P for construction stages i , which is expressed as follows:

[0051] P i = [x i1 , x i2 , …, x im , i = 1, 2, …, N

[0052] Initialize the particle velocity, which is expressed as follows:

[0053] v ij = Uniform(-v max , v max )

[0054]

[0055] In the formula, α is the proportionality coefficient;

[0056] 2) Calculate the fitness of the particle:

[0057] 3) Update the historical best position and global best position of the particle:

[0058] The update of the historical best position of the particle is expressed as: P i best = P i , if F(P i ) < F(P i best );

[0059] The update of the global best position is expressed as: P global = P i , if F(P i ) < F(P global );

[0060] 4) Update the particle velocity and position:

[0061] The particle velocity update is calculated by the following formula:

[0062]

[0063] The particle position update is calculated by the following formula:

[0064]

[0065] In the formula, $v_{ij}^t$ represents the velocity of the $i$-th particle in the $j$-th dimension after the $t$-th iteration, $w$ represents the inertia weight, $c_1$ represents the individual learning factor, $c_2$ represents the swarm learning factor, and $r_1$ and $r_2$ are randomly taken values in the interval $[0, 1]$. $x_{ij}^t$ represents the value of the position of the $i$-th particle in the $j$-th dimension at the $t$-th iteration;

[0066] 5) Constraint handling:

[0067] Particle velocity limit:

[0068] Particle position constraint:

[0069] 6) Termination condition: When the change in the global optimal fitness value satisfies , stop the iteration;

[0070] Step Ten: Output the optimization result

[0071] The optimized hanger force during the construction stage: Output the global optimal position $P$ of the particle swarm global :

[0072]

[0073] Verify the optimization effect: Input the optimized hanger force $X$ during the construction stage opt into the XGBoost model to predict the hanger force at the completed bridge stage:

[0074] $Y$ opt = f(X opt )

[0075] Calculate the optimized error: Compare the predicted hanger force $Y$ at the completed bridge stage after optimization opt with the ideal value $Y$ * :

[0076]

[0077] In the formula, $Y$ opt,j represents the predicted hanger force of the $j$-th hanger at the completed bridge stage after optimization;

[0078] Output result: Output the optimized hanger force $X$ during the construction stage opt , the predicted hanger force $Y$ at the completed bridge stage after optimization opt and the optimization error $F(X$ opt ).

[0079] Furthermore, in the above Step Nine, the maximum allowable value and the minimum allowable value respectively satisfy:

[0080]

[0081] In the formula, σ max represents the maximum allowable stress of the hanger material, and A j represents the cross-sectional area of the j-th hanger, and x safe represents the safety threshold.

[0082] Compared with the prior art, the beneficial effects of the present invention are as follows: By collecting the hanger forces in the construction stage and the hanger forces in the completed bridge stage of a suspension bridge, and combining the gradient boosting algorithm and the particle swarm optimization algorithm, the present invention proposes an intelligent optimization method for the hanger force in the construction stage of a suspension bridge with high precision and high efficiency. It can efficiently establish an accurate mapping relationship between the hanger forces in the construction stage and the hanger forces in the completed bridge stage, and quickly inverse the optimal hanger force distribution in the construction stage through a global optimization algorithm, comprehensively improving the optimization efficiency and precision, providing scientific support for the hanger tensioning in the construction stage of a suspension bridge, effectively improving the construction precision and the completed bridge quality of the bridge, and ensuring the structural safety and service performance of the bridge. BRIEF DESCRIPTION OF THE DRAWINGS

[0083] Figure 1 is a flowchart of the method of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0084] The following will clearly and completely describe the technical solutions in the present invention 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 invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0085] As Figure 1 shown, an intelligent optimization method for the hanger force in the construction stage of a suspension bridge based on gradient boosting includes the following steps:

[0086] Step 1: Data preparation

[0087] ① Input data:

[0088] Assume that there are m hangers in the construction stage of the suspension bridge, and the hanger force of each hanger is a feature, and the number of samples is n. For the i-th sample, the hanger force in the construction stage is expressed as:

[0089] X i = [x i1 , x i2 , …, x im

[0090] In the formula, x ij represents the hanger force in the construction stage of the j-th hanger of the i-th sample, where i = 1, 2, …, n and j = 1, 2, …, m. ​

[0091] ② Output data:

[0092] The hanger forces at the corresponding completed bridge stage are represented by a vector as follows:

[0093] Y i = [y i1 , y i2 , …, y im

[0094] where y ij represents the hanger force of the j-th hanger of the i-th sample at the completed bridge stage.

[0095] ③ Set representation of the dataset:

[0096] The input dataset of hanger forces during the construction stage is: X = {X1, X2, …, X n};

[0097] The output dataset of hanger forces at the completed bridge stage is: Y = {Y1, Y2, …, Y n}.

[0098] Step 2: Model definition and optimization objective

[0099] ① Model mapping relationship:

[0100] Establish a mapping function f to map the hanger forces during the construction stage to the hanger forces at the completed bridge stage, which is represented as follows:

[0101] f: X i → Y i

[0102] The goal of the model is to learn f to minimize the error between the predicted value and the true value Y i .

[0103] ② Optimization objective:

[0104] Use the squared error as the loss function L, which is represented as follows:

[0105]

[0106] where represents the predicted value of y ij .

[0107] ③ Objective formula:

[0108] Minimize the loss function as:

[0109] Step 3: Train the first tree

[0110] ① Initialize the predicted value:​

[0111] In the initial state, the model prediction value is the mean of the objective function, expressed as:

[0112] ② Calculate the initial residual:

[0113] The residual is expressed as the difference between the true value y ij and the current model prediction value :

[0114] Step Four: Determine the split point based on the gain

[0115] ① Gradient and second derivative:

[0116] Using the second-order Taylor expansion of the loss function, calculate the gradient and second derivative of each sample, expressed as follows:

[0117]

[0118] For the sample set D within the current node, there is:

[0119]

[0120] ② Gain formula:

[0121] For the split point v t , divide the data into the left child node D L and the right child node D R , there is:

[0122]

[0123] In the formula, G L and G R respectively represent the sum of gradients of the left and right child nodes, H L and H R respectively represent the sum of second derivatives of the left and right child nodes, λ represents the regularization parameter, and γ represents the split penalty term.

[0124] ③ Select the best split point:

[0125] Traverse all split points v t , and select the split point with the maximum gain:

[0126] Step Five: Update the leaf nodes

[0127] ① Output value of the leaf node:

[0128] For the leaf node, its output value is:

[0129]

[0130] ② Update the predicted value:

[0131] The predicted value of the model is updated to:

[0132]

[0133] In the formula, η represents the learning rate.

[0134] Step Six: Iterative training

[0135] ① Recalculate the residual:

[0136]

[0137] ② Termination condition:

[0138] Reach the maximum number of trees T:

[0139] κ = T

[0140] When the data scale is small (500 rows and below, 10 features and below), it is recommended that T be in the range of [50, 100); when the data scale is medium (more than 500 rows and less than 100,000 rows, more than 10 features and less than 100 features), it is recommended that T be in the range of [100 to 300); when the data scale is large (100,000 rows and above, 100 features and above), it is recommended that T be in the range of [300 to 500].

[0141] Or, the decrease in the loss function is less than the threshold ε:

[0142] |L (κ) - L (κ-1) | < ε

[0143] When the data scale is small, it is recommended that ε = 10 -3 ; when the data scale is medium, it is recommended that ε = 10 -4 ; when the data scale is large, it is recommended that ε = 10 -5 .

[0144] Step Seven: Predict new data

[0145] ① Input new data:

[0146] X new = [x new,1 , x new,2 , …, x new,m

[0147] ② Model prediction:

[0148] Use the cumulative result of all trees to calculate the new predicted value:

[0149]

[0150] In the formula, T κ (·) represents the prediction function of the κ-th tree, and T κ (X new ) represents the prediction result made by the κ-th tree on the input new data X new .

[0151] Step Eight: Error Analysis

[0152] Compare the hanger forces at the completed bridge stage predicted by the model and the ideal hanger forces Y * at the completed bridge stage, and calculate the error using the following formula:

[0153]

[0154] In the formula, is the hanger force of the j-th hanger at the completed bridge stage predicted, is the hanger force of the j-th hanger at the completed bridge stage of the ideal target.

[0155] If the error E is greater than the allowable threshold set by the project, then X new needs to be further adjusted so that the hanger forces at the completed bridge stage meet the ideal target.

[0156] Step Nine: Optimize the hanger forces during the construction stage using the particle swarm optimization algorithm

[0157] ① Definition of the optimization objective:

[0158] 1) Objective function:

[0159] Define that the optimized hanger forces X during the construction stage need to minimize the error between the predicted hanger forces at the completed bridge stage and the ideal target. The objective function is as follows:

[0160]

[0161] In the formula, f(X) j represents the hanger force of the j-th hanger at the completed bridge stage predicted by the model after optimization.

[0162] 2) Constraint conditions:

[0163] Define the upper and lower limit constraints for a single hanger:

[0164]

[0165] In the formula, and respectively represent the minimum allowable value and the maximum allowable value of the hanger force of the j-th hanger.

[0166] According to the design standards and engineering requirements, the hanger forces during the construction stage of each hanger need to meet the material strength limitations, so the maximum allowable value shall satisfy the following formula:

[0167]

[0168] In the formula, σ max represents the maximum allowable stress of the hanger material, and A j represents the cross-sectional area of the j-th hanger.

[0169] To ensure that the hanger is always in a tension state during the construction stage and avoid hanger relaxation or failure, a safety threshold lower than the minimum tension needs to be set. Therefore, the minimum allowable value shall satisfy the following formula:

[0170]

[0171] In the formula, x safe represents the safety threshold.

[0172] ② Particle Swarm Optimization Algorithm:

[0173] 1) Particle initialization:

[0174] Initialize N particles, and each particle represents a set of hanger forces P i during the construction stage, which is expressed as follows:

[0175] P i = [x i1 , x i2 , …, x im , i = 1, 2, …, N

[0176] Initialize the particle velocity, which is expressed as follows:

[0177] v ij = Uniform(-v max , v max )

[0178] v max is related to the position range of the particle (i.e., the upper and lower limits of the hanger force during the construction stage). To prevent the particle from moving too fast and jumping out of the search space, v max is determined by the following formula:

[0179]

[0180] In the formula, α is a proportionality coefficient with a value range of 0.1 to 1.

[0181] 2) Calculate the fitness of the particle:

[0182] The fitness value of each particle is calculated by the following formula:

[0183]

[0184] 3) Update the historical optimal position of the particle and the global optimal position:

[0185] The update of the historical optimal position of the particle is expressed as: P i best = P i , if F(P i ) < F(P i best );

[0186] The update of the global optimal position is expressed as: P global = P i , if F(P i ) < F(P global ).

[0187] 4) Update the particle velocity and position:

[0188] The particle velocity is updated by the following formula:

[0189]

[0190] The particle position is updated by the following formula:

[0191]

[0192] In the formula, represents the velocity of the i-th particle in the j-th dimension after the t-th iteration, w represents the inertia weight, which controls the influence of the current velocity of the particle in the next iteration. The recommended range of w is 0.4 - 0.9, c1 represents the individual learning factor, which controls the tendency of the particle to move towards its own historical optimal position, c2 represents the swarm learning factor, which controls the tendency of the particle to move towards the global optimal position. The recommended ranges of c1 and c2 are 1.5 - 2.5, r1 and r2 are randomly selected values in the range [0, 1], which are used to increase the randomness of the particle, represents the value of the position (hanger force during the construction stage) of the i-th particle in the j-th dimension at the t-th iteration, P i best is the historical optimal position of the i-th particle itself, and P global is the global optimal position.

[0193] 5) Constraint handling:

[0194] To avoid the particle velocity being too large or too small, it is usually necessary to constrain the velocity to prevent the particle from jumping out of the search space or falling into an invalid search. The particle velocity is limited by the following formula:

[0195]

[0196] For the particle position, the upper and lower limit constraints of the hanger force during the construction stage need to be satisfied:

[0197]

[0198] 6) Termination condition:

[0199] When the change in the global optimal fitness value satisfies stop the iteration.

[0200] Step Ten: Output the optimization result

[0201] ① Optimized hanger force during the construction stage:

[0202] Output the global optimal position P of the particle swarm global :

[0203]

[0204] ② Verify the optimization effect:

[0205] Input the optimized hanger force X during the construction stage opt into the xgboost model to predict the hanger force at the completed bridge stage:

[0206] Y opt = f(X opt )

[0207] ③ Calculate the optimized error:

[0208] Compare the predicted hanger force Y at the completed bridge stage after optimization opt with the ideal value Y * :

[0209]

[0210] In the formula, Y opt,j represents the hanger force at the completed bridge stage of the j-th hanger predicted after optimization.

[0211] ④ Output the result:

[0212] Output the optimized hanger force X during the construction stage opt , the predicted hanger force Y at the completed bridge stage after optimization opt and the optimization error F(X opt ).

[0213] Example

[0214] This example applies the method of the present invention to the Yangmingtan Bridge on the Songhua River, specifically as follows:

[0215] Step One: Data preparation

[0216] ① There are 30 suspension rods on the Yangmingtan Bridge of the Songhua River. A total of 100 samples are taken. Taking one of the samples as an example, the suspension rod forces during the construction stage are as follows:

[0217]

[0218] ② The corresponding suspension rod forces at the completed bridge stage are as follows:

[0219]

[0220] ③ Taking the first 5 out of 100 samples, the input dataset of suspension rod forces during the construction stage is shown as follows:

[0221]

[0222] The output dataset of suspension rod forces at the completed bridge stage is as follows:

[0223]

[0224] Step 2: Model Definition and Optimization Objective

[0225] ① Establish a mapping function f to map the suspension rod forces during the construction stage to those at the completed bridge stage, f: X i → Y i , and the goal of the model is to learn f to minimize the error between the predicted value and the true value Y i .

[0226] ② Use the squared error as the loss function L, which is expressed as:

[0227] ③ Minimize the loss function as:

[0228] Step 3: Train the First Tree

[0229] ① In the initial state, the model predicted value is the mean of the objective function:

[0230]

[0231] ② Taking the first 5 samples as an example, the residual is expressed as the difference between the true value y ij and the current model predicted value :

[0232]

[0233] Step 4: Determine the Splitting Point Based on the Gain

[0234] Find the best splitting point v based on the above residualst Let the splitting candidate point of the hanger force X be the middle value:

[0235]

[0236] Taking the first splitting point 881.43 as an example, the data is divided into the left child node D L and the right child node D R :

[0237]

[0238] In this embodiment, λ = 1 and γ = 0 are taken.

[0239] Splitting rule:

[0240] If the hanger force x ≤ 881.43, it enters the left child node; if x > 881.43, it enters the right child node.

[0241] Splitting result:

[0242] Left child node (hanger force x ≤ 881.43)

[0243] X L = [922.85], r L = [-296.59]

[0244] Right child node (hanger force x > 881.43)

[0245]

[0246] Calculate the gradient sum:

[0247] G L = ∑r L = -296.59, G R = ∑r R = -19816.1, H L = 1, H R = 29

[0248] Calculate the gain:

[0249]

[0250] Taking the first 5 samples as an example, traverse all splitting points v t and select the splitting point with the maximum gain:

[0251]

[0252] Step Five: Update the leaf nodes

[0253] ① For the leaf nodes, taking the first 5 samples as an example, their output values are:

[0254]

[0255] ② Update the predicted value and list one update:

[0256]

[0257] Take the learning rate as 0.01. Taking the first 5 samples as an example, the predicted values after update are as follows:

[0258]

[0259] Step Six: Iterative training

[0260] ① Recalculate the residual:

[0261] ② Termination condition (meeting either one): reaching the maximum number of trees T = 50, or the decrease in the loss function is less than the threshold ε = 10 -3 .

[0262] Step Seven: Predict new data

[0263] ① Input new data:

[0264]

[0265] ② Model prediction:

[0266] Use the accumulated results of all trees to calculate the new predicted value:

[0267]

[0268] Step Eight: Error analysis

[0269] The ideal hanger force at the completed bridge stage is:

[0270]

[0271] Compare the hanger force at the completed bridge stage predicted by the model and the ideal hanger force Y at the completed bridge stage * , and calculate the error

[0272] Step Nine: Optimize the hanger force during the construction stage using the particle swarm optimization algorithm

[0273] ① Define the optimization objective:

[0274] 1) Objective function:

[0275]

[0276] 2) Constraint conditions:

[0277] The force constraint of a single suspension rod is within the range of [500, 10000].

[0278] ② Particle Swarm Optimization Algorithm:

[0279] 1) Particle initialization:

[0280] The number of particle swarms is set to N = 50, the dimension of each particle is equal to the number of suspension rods m = 30, and the initial position P of each particle i is randomly generated within the same range as the suspension rod force constraint.

[0281] Take α = 1,

[0282] v max = 0.1×(10000 - 500) = 950

[0283] The initial velocity of each particle is randomly generated within the range of [-950, 950].

[0284] The positions and velocities of the first 5 particles are shown as follows:

[0285]

[0286] 2) Calculate the fitness of the particles:

[0287] The fitness values of the first 5 particles are as follows:

[0288]

[0289] 3) Update the historical optimal position and global optimal position of the particles:

[0290] The update of the historical optimal position of the particle is expressed as: P i best = P i , if F(P i ) < F(P i best );

[0291] The update of the global optimal position is expressed as: P global = P i , if F(P i ) < F(P global ).

[0292] 4) Update the particle velocity and position:

[0293] Taking the first 5 particles as an example, in this embodiment, one update is taken as an example. Since the initial position of the first-generation particles is their historical optimal position, therefore P i best = P i (0) , Pglobal = P i (0) , w = 0.6, c1 = c2 = 2.0, r1 = [0.3, 0.5, 0.6, 0.8, 0.2], r2 = [0.7, 0.4, 0.3, 0.5, 0.9].

[0294] After one iteration, the velocities of the second-generation particles are as follows:

[0295]

[0296] After updating the positions of the second-generation particles, they are as follows:

[0297]

[0298] 5) Constraint handling:

[0299] To prevent the particle velocities from being too large or too small, it is usually necessary to constrain the velocities to prevent the particles from jumping out of the search space or falling into ineffective searches. The particle velocities are limited by the following formula:

[0300]

[0301] For the particle positions, the upper and lower limit constraints of the hanger forces during the construction stage need to be satisfied:

[0302]

[0303] 6) Termination condition:

[0304] After calculation, after iteration:

[0305]

[0306] Stop iteration.

[0307] Step Ten: Output the optimization result

[0308] ① Output the global optimal position P of the particle swarm global :

[0309]

[0310] ② Input the optimized hanger force X during the construction stage opt into the xgboost model to predict the hanger force at the completed bridge stage:

[0311]

[0312] ③ Compare the predicted hanger force Y at the completed bridge stage after optimization opt with the ideal value Y * :

[0313]

[0314] ④ Output the optimized hanger force X during the construction stage opt , the predicted hanger force Y at the completed bridge stage after optimization opt and the optimization error F(X opt ).

[0315] For those skilled in the art, it is obvious that the present invention is not limited to the details of the above exemplary embodiments, and the present invention can be implemented in other form of device without departing from the spirit or basic characteristics of the present invention. Therefore, from any point of view, the embodiments should be regarded as exemplary and non-limiting. The scope of the present invention is defined by the appended claims rather than the above description. Therefore, all changes falling within the meaning and scope of the equivalent conditions of the claims are intended to be embraced within the present invention. Any reference signs in the claims should not be regarded as limiting the claimed claim.

[0316] In addition, it should be understood that although this specification is described according to the embodiments, not every embodiment only contains an independent technical solution. This narrative way of the specification is only for clarity. Those skilled in the art should regard the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. An intelligent optimization method for the hanger force during the construction stage of a suspension bridge based on gradient boosting, characterized in that: It includes the following steps: Step 1: Data preparation Suppose there are \(m\) suspenders in the construction stage of a suspension bridge, and the sample size is \(n\). For the \(i\)-th sample, the suspender force in the construction stage is expressed as: \(X\) i =\([x\) i1 ,x\) i2 ,\(\cdots,x\) im \), where \(x\) ij represents the suspender force of the \(j\)-th suspender in the \(i\)-th sample in the construction stage. The corresponding suspender force in the completed bridge stage is expressed as: \(Y\) i =\([y\) i1 ,y\) i2 ,\(\cdots,y\) im \), where \(y\) ij represents the suspender force of the \(j\)-th suspender in the \(i\)-th sample in the completed bridge stage. \(i = 1,2,\cdots,n\), \(j = 1,2,\cdots,m\). The input dataset of suspender forces in the construction stage is: \(X=\{X_1,X_2,\cdots,X\) n \}\), and the output dataset of suspender forces in the completed bridge stage is: \(Y=\{Y_1,Y_2,\cdots,Y\) n \}\); Step 2: Model definition and optimization objective Model mapping relationship: Establish a mapping function f to map the hanger forces in the construction stage to the hanger forces in the completed bridge stage. The goal of the model is to learn f to minimize the error between the predicted value and the true value Y i to the minimum; Optimization objective: Use the mean squared error as the loss function L, representing the predicted value of y ij ; Objective formula: Minimize the loss function which is Step 3: Train the first tree Initialize the predicted value: In the initial state, the model predicted value is the mean of the objective function, Calculate the initial residual: The residual is expressed as the true value y ij minus the current model prediction value : Step 4: Determine the splitting point based on the gain Gradient and second derivative: Using the second-order Taylor expansion of the loss function, calculate the gradient and second derivative of each sample, which are expressed as follows: For the sample set D within the current node, we have: Gain formula: For split point v t , divide the data into the left child node D L and the right child node D R , there is: where G L and G R represent the sums of gradients of the left and right child nodes respectively, H L and H R represent the sums of second-order derivatives of the left and right child nodes respectively, λ represents the regularization parameter, and γ represents the splitting penalty term; Select the best splitting point: Traverse all splitting points v t , and select the splitting point with the largest gain: Step 5: Update the leaf nodes Leaf node output value: For a leaf node, its output value is: Update predicted value: The model predicted value is updated to: η represents the learning rate; Step 6: Iterative training Recalculate the residual: Termination condition: Reach the maximum number of trees T or the decrease in the loss function is less than the threshold ε; Step 7: Predict new data Input new data: X new = [x new,1 , x new,2 , …, x new,m ; Model prediction: Use the cumulative result of all trees to calculate the new predicted value: where, T κ (·) represents the prediction function of the κ-th tree, T κ (X new ) represents the prediction result made by the κ-th tree on the input new data X new ; Step 8: Error analysis Compare the hanger forces at the completed bridge stage predicted by the model with the ideal hanger forces Y at the completed bridge stage * , and calculate the error using the following formula: In the formula, is the suspender force of the j-th suspender at the completed bridge stage predicted, is the suspender force of the j-th suspender at the completed bridge stage of the ideal target; If the error E is greater than the allowable threshold set by the project, then X is further adjusted new so that the hanger force in the completed bridge stage meets the ideal target; Step 9: Optimize the hanger force during the construction stage using the particle swarm optimization algorithm Optimization objective definition: 1) Objective function: Define that the optimized hanger force X during the construction stage needs to minimize the error between the predicted hanger force in the completed bridge stage and the ideal target. The objective function is as follows: where f(X) j represents the cable force of the j-th suspender at the completed bridge stage predicted by the optimized model; 2) Constraint conditions: Define the upper and lower limit constraints for a single hanger: In the formula, and respectively represent the minimum allowable value and the maximum allowable value of the force of the j-th suspension rod; Particle swarm optimization algorithm: 1) Particle initialization: Initialize N particles, where each particle represents a set of hanger forces P during the construction stage, expressed as follows: i , as follows: P i = [x i1 , x i2 , …, x im , i = 1, 2, …, N Initialize the particle velocity, which is expressed as follows: v ij = Uniform(-v max , v max ) In the formula, α is the proportionality coefficient; 2) Calculate the fitness of the particles: 3) Update the historical best position of the particle and the global best position: The update of the particle's historical optimal position is expressed as: P i best = P i , if F(P i ) < F(P i best ); The global optimal position update is expressed as: P global = P i , if F(P i ) < F(P global ); 4) Update the particle velocity and position: The particle velocity update is calculated by the following formula: The particle position update is calculated by the following formula: In the formula, represents the velocity of the i-th particle in the j-th dimension after the t-th iteration, w represents the inertia weight, c1 represents the individual learning factor, c2 represents the swarm learning factor, and r1 and r2 are randomly taken values in the interval [0, 1]. represents the value of the position of the i-th particle in the j-th dimension at the t-th iteration; 5) Constraint handling: Particle velocity limit: Particle position constraint: 6) Termination condition: When the change in the globally optimal fitness value satisfies stop the iteration; Step 10: Output the optimization result Optimized hanger force during construction stage: Output the global optimal position P of the particle swarm global : Verify the optimization effect: Input the optimized hanger force X during the construction stage opt into the XGBoost model to predict the hanger force at the completed bridge stage: Y opt = f(X opt ) Calculate the error after optimization: Compare the hanger forces Y at the completed bridge stage predicted after optimization opt with the ideal value Y * : where Y opt,j represents the cable force of the j-th hanger at the completed bridge stage predicted after optimization; Output result: Optimized hanger force X during construction stage opt , Predicted optimized hanger force Y at completed bridge stage opt and optimization error F(X opt ).

2. The intelligent optimization method for the hanger force at the construction stage of a suspension bridge based on gradient boosting according to claim 1, characterized in that: In the ninth step, the maximum allowable value and the minimum allowable value respectively satisfy: Where, σ max represents the maximum allowable stress of the suspension rod material, A j represents the cross-sectional area of the j-th suspension rod, and x safe represents the safety threshold.

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