Machine learning prediction method for p-y curve of single pile foundation in sandy soil

Abstract

The invention provides a sand single pile foundation p-y curve prediction method, which comprises the following steps of: firstly, integrating physical property parameters of sand and geometrical characteristics of a pile foundation as core input variables to train an advanced XGBoost prediction model; and then, inputting a series of soil body counterforces p on the target p-y curve and related input features thereof into the trained XGBoost model to obtain an accurate predicted value of p. Finally, a prediction result output by the XGBoost model and matched input characteristics of the prediction result are imported into the GPR model, and therefore a high-precision target p-y curve is generated. According to the method, a scientific and efficient solution is provided for p-y curve prediction of the sand single-pile foundation, and the method is particularly suitable for complex and variable uneven sand foundation scenes by means of excellent calculation speed and prediction precision. In addition, the technology shows strong generalization application capacity, and valuable guidance and reference are provided for design of the large-diameter single pile foundation in the field of offshore wind power.

Description

Technical Field

[0001] The present invention relates to the field of geotechnical engineering pile foundation design, and in particular to a machine learning prediction method for a PY curve of a single pile foundation in sand. Background Art

[0002] Monopile is a common foundation type for offshore wind turbines. In engineering design, the lateral bearing capacity of monopile foundation is usually evaluated using the nonlinear Winkler foundation beam (BNWF) model, which represents the soil resistance as a series of nonlinear springs that respond to the corresponding pile deformation. As a BNWF model, Figure 1 The Py curve method, shown here, has been widely used to quantify the lateral bearing capacity of piles under horizontal loads. This method characterizes the lateral pile-soil interaction along the soil depth by defining the relationship between the lateral soil resistance (p) and the lateral deflection (y) per unit length. The Py curve model recommended in existing specifications is based on field tests of small-diameter single pile foundations. As the size of offshore wind turbines increases, the calculation method recommended by the specifications will significantly overestimate the lateral bearing capacity of large-diameter single pile foundations.

[0003] In current research, researchers often modify the py curve model by adjusting the initial stiffness and skeleton function. However, these methods have poor generalization capabilities, especially in complex geological conditions. Furthermore, an increase in pile diameter can cause the failure mode of the pile foundation to transition from a "long, flexible pile" to a "short, rigid pile." Existing studies that extract the horizontal soil resistance p using the second-order derivative of cross-sectional bending moment with depth often include contributions from lateral friction, leading to deviations in the evaluation of the lateral response of large-diameter single piles. Summary of the Invention

[0004] In response to the technical problems existing in the above-mentioned background technology, the present invention proposes a machine learning prediction method for the py curve of a single pile foundation in sand. The method has a reasonable conception, good generalization ability and prediction accuracy, and not only provides a scientific and efficient solution for the py curve prediction of a single pile foundation in sand, but also, with its excellent calculation speed and prediction accuracy, is particularly suitable for complex and changeable uneven sand foundation scenarios.

[0005] To solve the above technical problems, the present invention provides a machine learning prediction method for the py curve of a single pile foundation in sand, which specifically includes the following steps:

[0006] 1) First, the physical properties of sand and the geometric characteristics of pile foundations are integrated as core input variables to train and optimize the XGBoost prediction model;

[0007] 2) Subsequently, a series of soil reaction forces p and their related core input variables on the py curve at the specified depth of the target single pile are input into the trained XGBoost prediction model to obtain the accurate prediction value of p;

[0008] 3) Finally, the predicted values ​​output by the XGBoost prediction model and its core input variables are imported into the GPR model to generate a high-precision target py curve.

[0009] The method for predicting the py curve of a single pile foundation in sandy soil by machine learning, wherein, in step 1), before training and optimizing the XGBoost prediction model, the original data set must first be systematically divided based on the specification requirements of the machine learning model training, specifically: the data set is divided into a training set, a validation set, and a test set in a ratio of 70%:15%:15% using a random sampling technique; the training set is used for iterative optimization of model parameters; the validation set is used for hyperparameter tuning and model selection; and the test set is used for performance evaluation of the optimized final model.

[0010] The single pile foundation py curve machine learning prediction method in the described sand soil, wherein the specific process of training and optimizing the XGBoost prediction model in the described step 1) is:

[0011] Starting from a set of initial hyperparameters, the objective function is modeled as a Gaussian process via Bayesian optimization, where each evaluated hyperparameter combination updates the model via Bayes’ theorem, which is expressed as:

[0012]

[0013] Where A represents the search domain of X, P represents the objective function, X and X+ correspond to the variable to be optimized and the optimal variable, respectively. To determine the next hyperparameter combination for evaluation, Bayesian optimization uses an acquisition function I(X), which is defined by the following formula (2):

[0014] I(X)=max{0,P t+1 (X)-P(X + )} (2);

[0015] Where t is the number of iterations; the acquisition function balances two conflicting objectives: exploration, which involves testing unknown regions of the hyperparameter space; and exploitation, which focuses on refining promising regions identified by the current model. Exploration ensures that the algorithm avoids getting stuck in local optima and uses exploration to fine-tune the search around high-performing regions. By iteratively updating the surrogate model and evaluating new hyperparameter combinations, Bayesian optimization converges to the optimal set of hyperparameters or terminates after a predefined computational budget is exhausted.

[0016] The machine learning prediction method for the py curve of a single pile foundation in sand, wherein the integration of the physical property parameters of the sand and the geometric characteristics of the pile foundation in step 1) is specifically: dimensionless processing is performed on the physical property parameters of the sand and the geometric parameter data of the single pile:

[0017]

[0018] In the above formula (3) and formula (4), D r is the relative density, is the critical friction angle, γ' is the effective gravity, γ w is the density of water, D is the pile diameter, L p is the burial depth, z is the depth, y is the lateral deformation, p is the lateral resistance of the soil, is the dimensionless number of deformation mechanism, z / D is the dimensionless number of depth, y / D is the dimensionless number of deformation, The ratio of severity.

[0019] The machine learning prediction method for the py curve of a single pile foundation in sand, wherein: in step 2), the six dimensionless core input variables D in formula (4) are preset at a specific depth of the sand. r 、 z / D, y / D and Imported into the trained XGBoost model, the XGBoost model will start from the root node according to the decision tree structure obtained by previous training, traverse the branches of the decision tree in sequence according to the value of the input variable, and finally reach the leaf node; each leaf node corresponds to a predicted value, and the predicted values ​​of all decision trees corresponding to the leaf nodes are added together to obtain the final predicted value of the soil reaction force p The calculation formula is as follows:

[0020]

[0021] where f k (x new ) represents the kth decision tree for input x new The predicted value of

[0022] Finally, we get the predicted values ​​under a series of different deformations on the target py curve.

[0023] The machine learning prediction method for the py curve of a single pile foundation in sandy soil, wherein the construction process of the GPR model in step 3) is:

[0024] 3.1) Given a training dataset D = {(x i ,y i )|i=1,...,n},input data X∈R D×n is called the design matrix, and y∈Rn A vector representing the expected output;

[0025] The different deformations of the specified depth of the pile foundation to be evaluated and other core input feature sets are imported into the trained and optimized XGBoost prediction model to generate a series of discrete Predicted value;

[0026] 3.2) Discrete the output of XGBoost prediction model The set of predicted values ​​and core input variables is the six dimensionless core input variables D in formula (4) r 、 z / D, y / D and As input, discrete The predicted values ​​and the above core input variable set are used as the training set to train the GPR model;

[0027] 3.3) Constructing Gaussian Process through RBF Kernel Function

[0028] The RBF kernel function is suitable for smooth py curves and optimizes the kernel parameters by maximizing the edge likelihood;

[0029] 3.4) Input the core input features used to train the GPR model again and convert the predicted value of GPR output into Connect to generate a smooth py curve.

[0030] The machine learning prediction method for the py curve of a single pile foundation in sandy soil, wherein the specific process of generating a high-precision target py curve in step 3) is as follows: after the predicted value output by the XGBoost model and its core input variables are imported into the GPR model, the GPR model assumes that the prior distribution of the function f is a multivariate normal distribution of the mean function μ(X) and the covariance matrix K(X,X) when there is no observation data:

[0031] f~N(μ,K) (6);

[0032] The covariance matrix K is composed of the kernel function k(x i ,x j ) definition, the RBF kernel function is:

[0033]

[0034] Reflects the similarity between input points; then, when integrating the observed data, constructs a joint distribution that combines the observed data (X, y) with the predicted point X * The posterior distribution of :

[0035]

[0036] in, is the observation noise variance, I is the identity matrix;

[0037] Then, the conditional probability is used to update the prior, that is, the prior is conditioned by the observed data y to obtain the predicted point X * The posterior distribution of :

[0038] f * |X * ,X,y~N(μ * ,∑ * ) (9);

[0039] The predicted value is the posterior mean μ * , variance Σ * Providing prediction confidence intervals, the RBF kernel function is suitable for smooth py curves, and the kernel parameters are optimized by maximizing the marginal likelihood to generate a high-precision target py curve.

[0040] The machine learning prediction method for the py curve of a single pile foundation in sandy soil, wherein the construction process of the XGBoost prediction model in step 1) is:

[0041] 1.1) Data Preparation and Feature Engineering

[0042] For a given dataset with n examples and m features The XGBoost prediction model uses K additive functions to predict the output:

[0043]

[0044] in, is the domain of the regression tree; q T represents the structure of each tree that maps examples to corresponding leaf indices; T is the number of leaves in the tree; each f k Corresponding to an independent tree structure q T and leaf weights w. Unlike decision trees, each regression tree contains a continuous score at each leaf, w i To represent the score on the i-th leaf;

[0045] Collect py curve data points, including 7 core variables: relative density D r , critical friction angle Deformation mechanism dimensionless number Depth dimensionless number z / D, deformation dimensionless number y / D, and weight ratio and dimensionless resistance Then the dataset was divided into 70% training set, 15% validation set, and 15% test set, and 3 independent curves were retained for generalization ability verification;

[0046] 1.2) XGBoost prediction objective function design

[0047] The objective function L consists of the loss function l and the regularization term Ω:

[0048]

[0049] In the above formula (11), l is a loss function used to measure the difference between the prediction and the target; the regularization term Ω controls the complexity of the model through the bias-variance trade-off, maintaining simplicity and prediction accuracy to prevent overfitting;

[0050] 1.3) Hyperparameter Optimization

[0051] Use Gaussian Process Regression (GPR) as a proxy model to search for key hyperparameters of XGBoost. Balance exploration and utilization through acquisition functions to efficiently approximate the optimal parameter combination.

[0052] 1.4) Tree model training and optimization

[0053] In XGBoost, the exact greedy algorithm is used for feature splitting of decision tree nodes. Specifically, when building a single decision tree, for the sample data of the current node to be split, all features and their possible split points are traversed, and the features and split points that can bring the greatest gain are selected for splitting:

[0054]

[0055] Where m is the number of data samples, is the predicted value of the i-th sample, is the observed value of the i-th sample, is the mean of the sample observations, is the average value of the sample prediction value; the termination condition is collaboratively controlled. If the evaluation index of the validation set does not improve after 50 consecutive rounds, the generation of new trees will be stopped, and the final model will be retained until the best iteration round;

[0056] The trained and optimized XGBoost model can be generated through the above steps 1.1)-1.4).

[0057] By adopting the above technical solution, the present invention has the following beneficial effects:

[0058] The present invention constructs a scientific and effective py curve prediction model for single pile foundation in sand, namely the XGBoost prediction model and the GPR model, which leverages the excellent ability of machine learning models to handle complex nonlinear problems.

[0059] The present invention has good generalization ability and prediction accuracy. It is only used for the lateral response of a single pile under the two deformation modes of "long flexible pile" and "short rigid pile", and can provide guidance for engineering design.

[0060] This invention not only provides a scientific and efficient solution for predicting the Py curve for sandy monopile foundations, but also, with its exceptional computational speed and prediction accuracy, is particularly applicable to complex and variable, uneven sandy foundation scenarios. Furthermore, this technology demonstrates strong generalizability, providing valuable guidance and reference for the design of large-diameter monopile foundations in the offshore wind power sector.

[0061] The present invention can solve the prediction problem of large-diameter single piles: it breaks through the limitations of the traditional API py curve model for small-diameter flexible piles and establishes a prediction method suitable for large-diameter rigid piles.

[0062] The present invention can enhance nonlinear modeling capabilities: it uses machine learning to capture the complex nonlinear relationships in pile-soil interactions, especially the stiffness changes and failure mode transitions caused by the diameter effect.

[0063] The present invention combines the XGBoost model's ability to quickly and accurately capture nonlinear relationships with the GPR model's smoothing properties for continuous space to achieve the prediction of the pile foundation py curve. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] In order to more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the specific embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0065] Figure 1 The process flow for estimating a single-stub py curve for an XGBoost model;

[0066] Figure 2 To demonstrate in detail the correspondence between observed values ​​and predicted values ​​when using this XGBoost-only model;

[0067] Figure 3 Fig. 3. Comparison between observed and predicted values ​​of the GPR model: (a) training set and (b) test set;

[0068] Figure 4 Fig.4. Comparison between observed and predicted values ​​of XGBoost+GPR model for all datasets value;

[0069] Figure 5 Fig.5. Comparison between observed data, XGBoost predicted data and XGBoost+GPR model predicted py curves;

[0070] Figure 6 Fig. 6. Comparison of the force-displacement curves obtained by back-calculation of the experimental and field tests with the predicted py curves;

[0071] Figure 7 Comparison of moment-depth curves obtained by inverse calculation of experimental and predicted py curves: (a) Qi et al. (2016); (b) Choo et al. (2016); (c) Reese et al. (1974);

[0072] Figure 8 Comparison of measured load-displacement curves and pile lateral responses predicted by the XGBoost+GPR model: (a) DM7; (b) DM4; (c) DM3; and (d) DL2;

[0073] Figure 9 Bending moment-depth curves of DL2 pile (a) when the ground bending moment is 10.8 MNm; and (b) when the ground bending moment is 34·1 MNm. DETAILED DESCRIPTION

[0074] The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the embodiments described are only some embodiments of the present invention, not all embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0075] The present invention will be further explained below with reference to specific embodiments.

[0076] like Figure 1 As shown, the present embodiment provides a machine learning prediction method for the py curve of a single pile foundation in sand, which mainly includes the following steps:

[0077] S100. First, the physical properties of sand and the geometric characteristics of pile foundation are integrated as core input variables to train and optimize the XGBoost prediction model.

[0078] Hyperparameter optimization in XGBoost (e.g., learning_rate, max_depth, subsamples) is critical to balancing model performance and complexity. Overfitting occurs when the model memorizes training noise (e.g., uncorrelated fluctuations) as meaningful patterns, thereby reducing generalization ability. Tuning alleviates this by adjusting regularization parameters (lambda, gamma), limiting tree depth, or reducing the ensemble size (n_estimators). Bayesian Optimization (BO) efficiently optimizes machine learning hyperparameters by guiding the search using a probabilistic surrogate model (typically Gaussian process regression). Starting from a set of initial hyperparameters, BO models the objective function as a Gaussian process, where each evaluated hyperparameter combination updates the model via Bayes' theorem, which can be expressed as:

[0079]

[0080] Where A represents the search domain of X, P represents the objective function, X and X+ correspond to the variable to be optimized and the optimal variable, respectively. In order to determine the next hyperparameter combination for evaluation, BO uses an acquisition function. The acquisition function I(X) is defined by the following equation (2):

[0081] I(X)=max{0,P t+1 (X)-P(X + )} (2);

[0082] Where t is the number of iterations; the acquisition function balances two conflicting objectives: exploration, which involves testing unknown areas of the hyperparameter space; and exploitation, which focuses on refining promising areas identified by the current model. Exploration ensures that the algorithm avoids getting stuck in local optimal solutions, while exploration allows fine-tuning the search around high-performance areas. By iteratively updating the surrogate model and evaluating new hyperparameter combinations, BO converges to the optimal hyperparameter set or terminates after a predefined computational budget is exhausted.

[0083] Based on the characteristics of the XGBoost prediction model and GPR machine learning models, different dataset construction strategies are adopted:

[0084] Before optimizing the XGBoost prediction model training, the original dataset needs to be systematically partitioned based on the standard requirements for machine learning model training. Specifically, the dataset is divided into training, validation, and test sets using a random sampling technique with a ratio of 70%:15%:15%. The training set (70%) is used for iterative optimization of model parameters; the validation set (15%) is used for hyperparameter tuning and model selection; and the test set (15%) is used for final model performance evaluation.

[0085] In view of the Bayesian property of the Gaussian process regression model, in this embodiment, the limited prediction results output by the XGBoost prediction model are combined with the corresponding input features to train the GPR model, and then the continuous deformation y is input into the trained GPR model to obtain the corresponding continuous resistance p.

[0086] The above integration of the physical properties of sand and the geometric characteristics of pile foundations is specifically to perform dimensionless processing on the physical properties of sand and the geometric parameters of single piles to improve the generalization ability of the training model:

[0087]

[0088] Among them, D in the above formula (3) and formula (4) r is the relative density, is the critical friction angle, γ' is the effective gravity, γ w is the density of water, D is the pile diameter, L p is the burial depth, z is the depth, y is the lateral deformation, p is the lateral resistance of the soil, is the dimensionless number of deformation mechanism, z / D is the dimensionless number of depth, y / D is the dimensionless number of deformation, The ratio of severity.

[0089] S200, the six dimensionless core input variables D in formula (4) at the preset specific depth of sand are r 、 z / D, y / D and Imported into the trained XGBoost prediction model, the XGBoost prediction model will start from the root node according to the decision tree structure obtained by previous training, traverse the branches of the decision tree in sequence according to the value of the input variable, and finally reach the leaf node; each leaf node corresponds to a prediction value, and the prediction values ​​of all decision trees corresponding to the leaf nodes are added together to obtain the final prediction value of the soil reaction force p The calculation formula is as follows:

[0090]

[0091] where f k (x new ) represents the kth decision tree for input x new The predicted value of

[0092] Finally, we get the predicted values ​​under a series of different deformations on the target py curve.

[0093] S300, a series of predicted values ​​output by the XGBoost prediction model By importing its corresponding deformation into the GPR model, a smooth and continuous py curve can be obtained;

[0094] The specific process of generating a high-precision target py curve is as follows: After importing the predicted values ​​output by the XGBoost prediction model and its core input variables into the GPR model, the GPR model assumes that the prior distribution of the function f is a multivariate normal distribution with the mean function μ(X) and the covariance matrix K(X,X) when there is no observation data:

[0095] f~N(μ,K) (6);

[0096] The covariance matrix K is composed of the kernel function k(x i ,x j ) definition, the RBF kernel function is:

[0097]

[0098] Reflects the similarity between input points; then, when integrating the observed data, constructs a joint distribution that combines the observed data (X, y) with the predicted point X * The posterior distribution of :

[0099]

[0100] in, is the observation noise variance, I is the identity matrix;

[0101] Then, the conditional probability is used to update the prior, that is, the prior is conditioned by the observed data y to obtain the predicted point X * The posterior distribution of :

[0102] f * |X * ,X,y~N(μ * ,∑ * ), (9);

[0103] The predicted value is the posterior mean μ * , variance Σ * Providing prediction confidence intervals, the RBF kernel function is suitable for smooth py curves, and the kernel parameters are optimized by maximizing the marginal likelihood to generate a high-precision target py curve.

[0104] The construction process of the above GPR model is:

[0105] 1) After the training of the XGBoost prediction model is completed, the different deformations of the specified depth of the pile foundation to be evaluated and other core input feature sets are imported into the trained and optimized XGBoost prediction model to generate a series of discrete Predicted value.

[0106] 2) Discrete the output of XGBoost prediction model The set of predicted values ​​and core input variables is the six dimensionless core input variables D in formula (4) r 、 z / D, y / D and As input, discrete The predicted values ​​and the above core input variable set are used as the training set to train the GPR model;

[0107] 3) Constructing Gaussian process through RBF kernel function: RBF kernel function is suitable for smooth py curve, and kernel parameters (such as length scale l, noise variance) are optimized by maximizing edge likelihood. ).

[0108] 4) Input the core input features used to train the GPR model again and convert the predicted value of GPR output into Connect to generate a smooth py curve.

[0109] The XGBoost+GPR model in the above steps S200 and S300, Figure 1 Shown is the forecast flow chart, which can be divided into the following parts;

[0110] a) Divide the dataset

[0111] To ensure accurate prediction results, model training requires a database containing a representative range of samples. Table 1 provides the database used in this study. It contains 221 py curve data points (out of a total of 2554 data points) from 19 representative studies on the lateral response of piles. Among them, 43 py curves were obtained from three sources using a detailed numerical validation model. The dataset is divided into three different subsets: training set, validation set, and unseen test set. The training set and validation set account for 70% and 15% of the total dataset, respectively, and play a key role in adjusting the weights and hyperparameters of the model during the training process. The test set also accounts for 15% of the total dataset and is used to evaluate the trained model.

[0112] Table 1. Characteristic parameter ranges of pile lateral response tests

[0113]

[0114]

[0115]

[0116] b) eXtreme Gradient Boosting (XGBoost)

[0117] The core concept of XGBoost is to build multiple trees, where each new tree is designed to correct the errors of the previous trees, thereby gradually improving the performance of the model.

[0118] For a given dataset with n examples and m features The tree ensemble model, i.e., the XGBoost prediction model, uses K additive functions to predict the output.

[0119]

[0120] in is the domain of the regression tree. Here q T represents the structure of each tree that maps examples to corresponding leaf indices. T is the number of leaves in the tree. Each f k Corresponding to an independent tree structure q T and leaf weights w. Unlike decision trees, each regression tree contains a continuous score at each leaf, using w i To represent the score on the i-th leaf.

[0121] 2,554 py curve data points were collected from 19 studies, including 7 core variables: relative density D r , critical friction angle Deformation mechanism dimensionless number Depth dimensionless number z / D, deformation dimensionless number y / D, and weight ratio and dimensionless resistance The dataset was then divided into 70% training set, 15% validation set, and 15% test set, and 3 independent curves were retained for generalization ability verification.

[0122] The objective function L consists of the loss function l and the regularization term Ω:

[0123]

[0124] Here l is a loss function that measures the difference between the prediction and the target; the regularization term Ω controls the complexity of the model through the bias-variance trade-off, maintaining simplicity and prediction accuracy, which helps prevent overfitting.

[0125] Gaussian process regression (GPR) is used as a proxy model to search for XGBoost key hyperparameters. The acquisition function is used to balance exploration and utilization to efficiently approximate the optimal parameter combination. Specifically, the validation set performance evaluation index CC (Formula (12)) is used as the objective function of Bayesian optimization, and initial points are randomly generated in the preset space to efficiently approximate the optimal parameter combination. In XGBoost, the exact greedy algorithm is used for feature splitting of decision tree nodes. Specifically, when constructing a single decision tree, all features and possible split points of the sample data of the current node to be split are traversed, and the features and split points that can bring the maximum gain are selected for splitting:

[0126]

[0127] Where m is the number of data samples, is the predicted value of the i-th sample, is the observed value of the i-th sample, is the mean of the sample observations, is the average value of the sample prediction value; the termination condition is collaboratively controlled. If the evaluation index of the validation set does not improve after 50 consecutive rounds, the generation of new trees will be stopped, and the final model will be retained until the best iteration round; the above process can generate the trained and optimized XGBoost prediction model.

[0128] c) Gaussian Process Regression (GPR)

[0129] GPR is a probabilistic and nonparametric supervised learning method that aims to generalize nonlinear and complex function mappings hidden in the dataset. By utilizing kernel functions, GPR has been shown to be very effective in processing nonlinear data. Given a training dataset D = {(x i ,y i )|i=1,...,n},input data X∈R D×n is called the design matrix, and y∈R n A vector representing the expected output. The main assumptions of Gaussian Process Regression (GPR) are:

[0130] y=f(x)+ε (13);

[0131] in It represents the mean squared deviation of an observed output from its expected value due to measurement error or inherent randomness in the data. In the GPR method, the n observations in the dataset are treated as samples drawn from a multivariate Gaussian distribution. In Gaussian process regression, the function f(x) is assumed to be distributed as a Gaussian process.

[0132] f(x)~GP(m(x),k(x,x′))(14);

[0133] A Gaussian process (GP) is a distribution over functions, defined by a mean and covariance function. The mean function m(x) reflects the expected function value at input x. The function k(x, x') is often referred to as the kernel of the Gaussian process. The selection of an appropriate kernel is based on assumptions such as smoothness and the likely patterns expected in the data. In this study, the primary kernel used was the radial basis function (RBF) kernel combined with a white noise kernel.

[0134]

[0135] The radial basis function provides an expressive kernel to simulate smooth and stationary functions. The hyperparameters in the kernel function are θ = {l,σ f ,σ n}, expressed as length scale l, signal variance σ f and the noise level σ n The length scale l describes the function σ f The smaller l is, the faster the function changes, while the larger l is, the smoother the prediction. Signal variance σ f is a scaling factor that controls the variation of the predicted value relative to the mean. n specifies the expected amount of noise in the observations (or training data).

[0136] Evaluation of XGBoost prediction model and GPR model:

[0137] Figure 2 The correspondence between the observed and predicted values ​​when using this XGBoost-only prediction model is demonstrated in detail. The proposed XGBoost prediction model shows excellent performance indicators on the training set: the root mean square error (RMSE) is as low as 0.172, the scattering index (SI) is 0.019, reflecting the stability of the model prediction, and the correlation coefficient (CC) reaches 0.999, which is almost perfect. In addition, the model also performs well on the test set, with an RMSE of 1.034. Although this is slightly higher than the performance of the training set, it is still at a relatively low level; the SI is 0.114, close to 0, and the CC is 0.990. These results together confirm that the XGBoost prediction model shows excellent accuracy and generalization ability in evaluating the soil reaction p under specific deformation at a specified depth for different soils.

[0138] Figure 3Using only a Gaussian process regression (GPR) model, we provide a visualization of the correspondence between observed and predicted values. For the training set, the GPR model achieved satisfactory performance metrics: a root mean square error (RMSE) of 9.160, a scatter index (SI) of 0.971, and a correlation coefficient (CC) of 0.329. However, when applied to an independent test set, the model's predictive performance deteriorated. Evaluation results on the test set showed an increase in RMSE to 8.830, an SI value of 1.044, and a decrease in CC to 0.355. These metrics indicate that the model's predictions for unseen data deviate significantly from the actual observed values, reflecting a lack of generalization ability.

[0139] The output of the XGBoost prediction model is the dimensionless reaction force These scattered points cannot be directly connected to form a smooth py curve. The GPR model can further process the scattered point set by taking advantage of its ability to regress and smooth the curve based on a limited number of data points. Figure 1 The two models shown in the flowchart in the figure, the hybrid model retains the XGBoost prediction model's ability to accurately predict the soil reaction force p under specific conditions. Figure 4 The comparison between the predicted and observed values ​​of the hybrid model for the entire dataset is shown. The RMSE of the hybrid model is 0.445, SI is 0.051, and CC is 0.999. In the hybrid model, the GPR component uses the output of the trained XGBoost prediction model as input, as shown in Figure 4 Therefore, the comparative analysis did not distinguish between the training and test datasets. The three curves from Qi et al. (2016), Choo and Kim (2016), and Wang et al. (2021) were excluded from the training and testing processes to evaluate the generalization performance of the XGBoost+GPR model. Figure 5 The comparison results in Figure 3 show that although the output points of the XGBoost prediction model are closely distributed around the corresponding py curve, directly connecting these points does not form a smooth curve consistent with the physical mechanism. This phenomenon occurs because it is impossible to ensure that the sampling range and density of py curves from different sources in the training set are exactly the same. In contrast, the continuous smooth curve output by the XGBoost+GPR model closely matches the observed py curve.

[0140] By using the XGBoost+GPR model, a series of py curves distributed along the pile can be obtained, which can then be used to inversely calculate the force-displacement and moment-depth curves of a single pile, thereby verifying the generalization ability of the prediction model. The py data predicted by the XGBoost+GPR model can be applied to the beam model to calculate the lateral response of the pile. Figure 6 and Figure 7As shown in Figure 3, the lateral pile response backcalculated from the predicted py curve is in good agreement with the test results, indicating that the trained XGBoost+GPR model can be used to evaluate the lateral bearing capacity of piles.

[0141] The PISA (Pile-Soil Analysis) project combined ground characterization, field testing, and computational analysis to develop a new design model for large-diameter monopiles used as foundations for offshore wind turbines. Fourteen open-end steel piles were installed at the Dunkirk site and subjected to lateral loads. Previous studies did not involve the construction or training of a predictive model. This study only compared four typical cases mentioned in Taborda et al. (2020). Figure 8 and Figure 9 A comparison between the measured lateral responses and the predicted model responses for four different pile diameters is shown. The comparison shows that although the predictions of the lateral pile responses by the XGBoost+GPR model show slight deviations from the field test results, they are closely related to the numerical model proposed by Taborda et al. (2020) or the optimization model proposed by Burd et al. (2019), respectively. These deviations may be due to the inherent complexity of the site geological conditions and the inevitable simplifications in the modeling and simulation process. Despite these limitations, the trained prediction models have shown sufficient reliability in guiding engineering practice. Compared with the API (2014) method, which is limited to design under homogeneous sand conditions, the proposed model provides a more flexible framework that can effectively solve complex problems.

[0142] The present invention has a reasonable concept, good generalization ability and prediction accuracy. It not only provides a scientific and efficient solution for the prediction of the PY curve of a single pile foundation in sand, but also is particularly suitable for complex and changeable uneven sand foundation scenarios due to its excellent calculation speed and prediction accuracy.

[0143] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A machine learning prediction method for the py curve of a single pile foundation in sandy soil, characterized in that: The specific steps include: 1) First, the physical properties of sand and the geometric characteristics of pile foundations are integrated as core input variables to train and optimize the XGBoost prediction model; 2) Subsequently, a series of soil reaction forces p and their related core input variables on the py curve at the specified depth of the target single pile are input into the trained XGBoost prediction model to obtain the accurate prediction value of p; 3) Finally, the predicted values ​​output by the XGBoost prediction model and its core input variables are imported into the GPR model to generate a high-precision target py curve.

2. The single pile foundation py curve machine learning prediction method in sandy soil as claimed in claim 1, wherein In step 1), before training and optimizing the XGBoost prediction model, the original data set must be systematically divided based on the standard requirements for machine learning model training. Specifically, the data set is divided into a training set, a validation set, and a test set in a ratio of 70%:15%:15% using a random sampling technique; the training set is used for iterative optimization of model parameters; the validation set is used for hyperparameter tuning and model selection; and the test set is used for performance evaluation of the optimized final model.

3. The single pile foundation py curve machine learning prediction method in sandy soil as claimed in claim 1, wherein The specific process of training and optimizing the XGBoost prediction model in step 1) is as follows: Starting from a set of initial hyperparameters, the objective function is modeled as a Gaussian process via Bayesian optimization, where each evaluated hyperparameter combination updates the model via Bayes’ theorem, which is expressed as: Where A represents the search domain of X, P represents the objective function, X and X+ correspond to the variable to be optimized and the optimal variable, respectively. To determine the next hyperparameter combination for evaluation, Bayesian optimization uses an acquisition function I(X), which is defined by the following formula (2): I(X)=max{0,P t+1 (X)-P(X + )} (2); Where t is the number of iterations; the acquisition function balances two conflicting objectives: exploration, which involves testing unknown regions of the hyperparameter space; and exploitation, which focuses on refining promising regions identified by the current model. Exploration ensures that the algorithm avoids getting stuck in local optima and uses exploration to fine-tune the search around high-performing regions. By iteratively updating the surrogate model and evaluating new hyperparameter combinations, Bayesian optimization converges to the optimal set of hyperparameters or terminates after a predefined computational budget is exhausted.

4. The single pile foundation py curve machine learning prediction method in sandy soil as claimed in claim 1, wherein The integration of the physical property parameters of the sand and the geometric characteristics of the pile foundation in step 1) is specifically: dimensionless processing is performed on the physical property parameters of the sand and the geometric parameter data of the single pile: In the above formula (3) and formula (4), D r is the relative density, is the critical friction angle, γ' is the effective gravity, γ w is the density of water, D is the pile diameter, L p is the burial depth, z is the depth, y is the lateral deformation, p is the lateral resistance of the soil, is the dimensionless number of deformation mechanism, z / D is the dimensionless number of depth, y / D is the dimensionless number of deformation, The ratio of severity.

5. The method for predicting the py curve of a single pile foundation in sandy soil according to claim 4, wherein: In step 2), the six dimensionless core input variables D in formula (4) are set at a specific depth of the preset sand. r 、 z / D, y / D and Imported into the trained XGBoost model, the XGBoost model will start from the root node according to the decision tree structure obtained by previous training, traverse the branches of the decision tree in sequence according to the value of the input variable, and finally reach the leaf node; each leaf node corresponds to a predicted value, and the predicted values ​​of all decision trees corresponding to the leaf nodes are added together to obtain the final predicted value of the soil reaction force p The calculation formula is as follows: where f k (x new ) represents the kth decision tree for input x new The predicted value of Finally, we get the predicted values ​​under a series of different deformations on the target py curve.

6. The method for predicting the py curve of a single pile foundation in sandy soil according to claim 4, wherein: The construction process of the GPR model in step 3) is as follows: 3.1) Given a training dataset D = {(x i ,y i )|i=1,...,n},input data X∈R D×n is called the design matrix, and y∈R n A vector representing the expected output; The different deformations of the specified depth of the pile foundation to be evaluated and other core input feature sets are imported into the trained and optimized XGBoost prediction model to generate a series of discrete Predicted value; 3.2) Discrete the output of XGBoost prediction model The set of predicted values ​​and core input variables is the six dimensionless core input variables D in formula (4) r 、 z / D, y / D and As input, discrete The predicted values ​​and the above core input variable set are used as the training set to train the GPR model; 3.3) Constructing Gaussian Process through RBF Kernel Function The RBF kernel function is suitable for smooth py curves and optimizes the kernel parameters by maximizing the edge likelihood; 3.4) Input the core input features used to train the GPR model again and convert the predicted value of GPR output into Connect to generate a smooth py curve.

7. The single pile foundation py curve machine learning prediction method in sandy soil as claimed in claim 1, characterized in that The specific process of generating a high-precision target py curve in step 3) is as follows: after importing the predicted value output by the XGBoost model and its core input variables into the GPR model, the GPR model assumes that the prior distribution of the function f is a multivariate normal distribution of the mean function μ(X) and the covariance matrix K(X,X) when there is no observation data: f~N(μ,K)(6); The covariance matrix K is composed of the kernel function k(x i ,x j ) definition, the RBF kernel function is: Reflects the similarity between input points; Then, when integrating the observed data, a joint distribution is constructed, which combines the observed data (X, y) with the predicted point X * The posterior distribution of : in, is the observation noise variance, I is the identity matrix; Then, the conditional probability is used to update the prior, that is, the prior is conditioned by the observed data y to obtain the predicted point X * The posterior distribution of : f * |X * ,X,y~N(μ * ,∑ * ) (9)? The predicted value is the posterior mean μ * , variance Σ * Providing prediction confidence intervals, the RBF kernel function is suitable for smooth py curves, and the kernel parameters are optimized by maximizing the marginal likelihood to generate a high-precision target py curve.

8. The method for predicting the py curve of a single pile foundation in sandy soil according to claim 1, wherein: The construction process of the XGBoost prediction model in step 1) is as follows: 1.1) Data Preparation and Feature Engineering For a given dataset with n examples and m features The XGBoost prediction model uses K additive functions to predict the output: in, is the domain of the regression tree; q T represents the structure of each tree that maps examples to corresponding leaf indices; T is the number of leaves in the tree; each f k Corresponding to an independent tree structure q T and leaf weights w. Unlike decision trees, each regression tree contains a continuous score at each leaf, w i To represent the score on the i-th leaf; Collect py curve data points, including 7 core variables: relative density D r , critical friction angle Deformation mechanism dimensionless number Depth dimensionless number z / D, deformation dimensionless number y / D, and weight ratio and dimensionless resistance Then the dataset was divided into 70% training set, 15% validation set, and 15% test set, and 3 independent curves were retained for generalization ability verification; 1.2) XGBoost prediction objective function design The objective function L consists of the loss function l and the regularization term Ω: In the above formula (11), l is a loss function used to measure the difference between the prediction and the target; the regularization term Ω controls the complexity of the model through the bias-variance trade-off, maintaining simplicity and prediction accuracy to prevent overfitting; 1.3) Hyperparameter Optimization Use Gaussian Process Regression (GPR) as a proxy model to search for key hyperparameters of XGBoost. Balance exploration and utilization through acquisition functions to efficiently approximate the optimal parameter combination. 1.4) Tree model training and optimization In XGBoost, the exact greedy algorithm is used for feature splitting of decision tree nodes. Specifically, when building a single decision tree, for the sample data of the current node to be split, all features and their possible split points are traversed, and the features and split points that can bring the greatest gain are selected for splitting: Where m is the number of data samples, is the predicted value of the i-th sample, is the observed value of the i-th sample, is the mean of the sample observations, is the average value of the sample prediction value; the termination condition is collaboratively controlled. If the evaluation index of the validation set does not improve after 50 consecutive rounds, the generation of new trees will be stopped, and the final model will be retained until the best iteration round; The trained and optimized XGBoost model can be generated through the above steps 1.1)-1.4).

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