Sand soil single pile foundation p-y curve machine learning prediction method

Abstract

The application provides a sandy soil single pile foundation p-y curve prediction method, which is firstly, the physical property parameters of sandy soil and the geometric characteristics of pile foundation are integrated as core input variables to train an advanced XGBoost prediction model. Then, a series of soil reaction forces p on the target p-y curve and the related input characteristics are input into the trained XGBoost model to obtain the accurate prediction value of p. Finally, the prediction results output by the XGBoost model and the matching input characteristics are introduced into the GPR model to generate a high-precision target p-y curve. The application not only provides a scientific and efficient solution for the p-y curve prediction of sandy soil single pile foundation, but also is especially suitable for complex and variable uneven sandy soil foundation scenes due to its excellent calculation speed and prediction accuracy. In addition, the technology shows strong general application ability and provides valuable guidance and reference for the design of large-diameter single pile foundation in the offshore wind power field.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of geotechnical engineering pile foundation design, and particularly relates to a p-y curve machine learning prediction method for single pile foundation in sand. BACKGROUND

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

[0003] In the current study, researchers often modify the p-y curve model by adjusting the initial stiffness and the skeleton function. However, these methods have poor generalization ability, especially for complex geological conditions. At the same time, the increase in pile diameter will also make the failure mode of the pile foundation transition from "long and soft pile" to "short and rigid pile". The method of extracting horizontal soil resistance p by the second derivative of sectional bending distance with depth in existing research will have the contribution of side friction resistance, which will cause deviation in evaluating the lateral response of large-diameter single pile. SUMMARY

[0004] To solve the technical problems in the above background art, the present application provides a p-y curve machine learning prediction method for single pile foundation in sand, which has reasonable concept, good generalization ability and prediction accuracy. It not only provides a scientific and efficient solution for p-y curve prediction of single pile foundation in sand, but also is especially suitable for complex and variable non-uniform sand foundation scenarios due to its excellent calculation speed and prediction accuracy.

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

[0006] 1) First, integrate the physical property parameters of sand and the geometric characteristics of pile foundation as core input variables to train and optimize the XGBoost prediction model;

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

[0008] 3) Finally, the predicted value and its core input variables output by the XGBoost prediction model are imported into the GPR model, thereby generating a high-precision target p-y curve.

[0009] The machine learning prediction method for the p-y curve of a single pile foundation in sandy soil, wherein before training and optimizing the XGBoost prediction model, the original data set is systematically divided based on the specification requirements for model training, specifically: using random sampling technology to divide the data set into training set, validation set and test set in the ratio of 70:15:15; the training set is used for model parameter iterative optimization; the validation set is used for hyperparameter tuning and model selection; the test set is used for performance evaluation of the optimized final model.

[0010] The machine learning prediction method for the p-y curve of a single pile foundation in sandy soil, wherein the specific process of training and optimizing the XGBoost prediction model in step 1) is as follows:

[0011] Starting from a set of initial hyperparameters, the objective function is modeled as a Gaussian process through Bayesian optimization, where each evaluated hyperparameter combination updates the model through Bayesian theorem, and the process is represented as:

[0012]

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

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

[0015] Where t represents the number of iterations; the acquisition function balances two conflicting objectives: exploration, which involves testing unknown areas of the hyperparameter space; exploitation, which focuses on refining promising areas identified by the current model; exploration ensures that the algorithm avoids local optimal solutions and fine-tunes the search around high-performance areas; through iterative updates of the surrogate model and evaluation of new hyperparameter combinations, Bayesian optimization converges to the optimal hyperparameter set or terminates after the predefined computational budget is exhausted.

[0016] The sand soil single pile foundation p-y curve machine learning prediction method, wherein the step 1) integrates the physical property parameters of the sand soil and the geometric characteristics of the pile foundation, and specifically is: dimensionless processing is performed on the sand soil physical property parameter and single pile geometric parameter data:

[0017]

[0018] 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 gravity of water, D is the pile diameter, L p is the buried depth, z is the depth, y is the lateral deformation, p is the lateral resistance of the soil body, is the dimensionless number of deformation mechanism, z / D is the depth dimensionless number, y / D is the deformation dimensionless number, is the gravity ratio.

[0019] The sand soil single pile foundation p-y curve machine learning prediction method, wherein: in the step 2), six dimensionless core input variables D r , z / D, y / D and in formula (4) at a predetermined depth position of the sand soil are introduced into the trained XGBoost model, the XGBoost model will start from the root node according to the decision tree structure obtained by previous training, and according to the value of the input variable, the branches of the decision tree are traversed in turn, and finally the leaf node is reached; each leaf node corresponds to a predicted value, and the predicted values of the leaf nodes corresponding to all decision trees are added 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 predicted value of the kth decision tree for the input x new ;

[0022] The final predicted value of a series of different deformations on the target p-y curve is obtained

[0023] The sand soil single pile foundation p-y curve machine learning prediction method, wherein the construction process of the GPR model in the step 3) is:

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

[0025] The different deformations of the pile foundation at a specified depth to be evaluated, along with other core input feature sets, are imported into the trained and optimized XGBoost prediction model to generate a series of discrete values ​​that correspond one-to-one with the core input feature values. Predicted value;

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

[0027] 3.3) Constructing a Gaussian process using the RBF kernel function

[0028] The RBF kernel function is suitable for smooth Py curves, and optimizes the kernel parameters by maximizing the marginal likelihood.

[0029] 3.4) Re-input the core input features used to train the GPR model, and output the predicted values ​​of the GPR model. Connecting the lines will generate a smooth Python 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 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.

[0031] f~N(μ,K) (6);

[0032] The covariance matrix K is determined by the kernel function k(x) i ,x j The RBF kernel function is defined as follows:

[0033]

[0034] This reflects the similarity between input points; subsequently, when integrating the observation data, a joint distribution is constructed, linking the observation data (X, y) with the point to be predicted X. * The posterior distribution:

[0035]

[0036] in, I is an identity matrix;

[0037] Then the prior is updated using conditional probability, i.e. the prior is conditioned on the observed data y to obtain the posterior distribution of the prediction point X * :

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

[0039] The prediction value takes the posterior mean μ * , and the variance Σ * provides the prediction confidence interval. The RBF kernel function is suitable for smooth p-y curves, and the kernel parameters are optimized by maximizing the marginal likelihood, thereby generating high-precision target p-y curves.

[0040] The sand soil single pile foundation p-y curve machine learning prediction method, wherein the construction process of the XGBoost prediction model in step 1) is as follows:

[0041] 1.1) Data preparation and feature engineering

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

[0043]

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

[0045] Collect p-y curve data points, including 7 core variables: relative density D r , critical friction angle dimensionless number of deformation mechanism dimensionless number of depth z / D, dimensionless number of deformation y / D, ratio of specific gravity and dimensionless resistance Then divide it into 70% training set, 15% validation set and 15% test set, and reserve 3 independent curves for generalization ability verification;

[0046] 1.2) XGBoost prediction objective function design

[0047] The objective function L is composed of a loss function l and a regularization term Ω:

[0048]

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

[0050] 1.3) Hyperparameter optimization

[0051] Gaussian process regression GPR is used as a surrogate model to search for key hyperparameters of XGBoost; the acquisition function balances exploration and utilization 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 constructing 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 feature and split point that can bring the largest 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 average of the sample observed values, is the average of the sample predicted values; the termination condition is cooperatively controlled, if the evaluation index of the validation set has not improved for 50 consecutive rounds, the generation of new trees is stopped, and the final model is retained to the best iteration round;

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

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

[0058] The present application constructs a scientific and effective p-y curve prediction model for single pile foundation in sandy soil, namely XGBoost prediction model and GPR model, which takes advantage of the excellent ability of machine learning model to handle complex nonlinear problems.

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

[0060] The present application not only provides a scientific and efficient solution for the prediction of p-y curves of sandy soil single pile foundation, but also is particularly suitable for complex and variable uneven sandy soil foundation scenarios due to its excellent calculation speed and prediction accuracy. In addition, the technology shows strong general application ability, providing valuable guidance and reference for the design of large-diameter single pile foundation in the offshore wind power field.

[0061] The present application can solve the prediction problem of large-diameter single pile: break through the limitation of traditional API p-y curve model in small-diameter flexible pile, and establish a prediction method suitable for large-diameter rigid pile.

[0062] The present application can improve the nonlinear modeling capability: through machine learning to capture the complex nonlinear relationship in pile-soil interaction, especially the stiffness change and failure mode transition caused by diameter effect.

[0063] The present application combines the ability of XGBoost model to quickly and accurately capture nonlinear relationship with the smoothing characteristics of GPR model in continuous space to realize the prediction of pile foundation p-y curve. BRIEF DESCRIPTION OF DRAWINGS

[0064] In order to more clearly illustrate the specific embodiments of the present application or the technical solutions in the prior art, the drawings needed in the description of the specific embodiments or the prior art will be briefly introduced. Obviously, the drawings in the following description are some embodiments of the present application, and for those skilled in the art, other drawings can also be obtained without creative labor on the basis of these drawings.

[0065] Figure 1 Process flow for XGBoost model to estimate single pile p-y curve;

[0066] Figure 2 To demonstrate the correspondence between the observed value and the predicted value when using only the XGBoost model;

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

[0068] Figure 4 Fig. 4. Comparison of observed value and predicted value of XGBoost+GPR model of all data sets Value;

[0069] Figure 5 Fig. 5. Comparison between observed data, XGBoost predicted data and XGBoost+GPR model predicted p-y curve;

[0070] Figure 6 Fig. 6. Comparison of the test and field test with the predicted p-y curve back-calculated force-displacement curves;

[0071] Figure 7 Comparison of the test and predicted p-y curve back-calculated moment-depth curves: (a) Qi et al. (2016); (b) Choo et al. (2016); (c) Reese et al. (1974);

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

[0073] Figure 9 Bending moment-depth curves of DL2 pile (a) Ground bending moment is 10.8 MNm; and (b) Ground bending moment is 34.1 MNm. DETAILED DESCRIPTION

[0074] The technical solutions of the present application will be described clearly and completely below in conjunction with the drawings. Obviously, the described embodiments are part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative labor fall within the scope of protection of the present application.

[0075] The present application will be further explained and described below in conjunction with specific embodiments.

[0076] As shown in the embodiment, the sand soil single pile foundation p-y curve machine learning prediction method provided by the embodiment mainly includes the following steps: Figure 1 S100, first, integrate the physical property parameters of the sand soil and the geometric characteristics of the pile foundation as core input variables to train and optimize the XGBoost prediction model;

[0077]

[0078] ​Hyperparameter optimization in XGBoost (e.g., learning_rate, max_depth, subsample) is critical to balance model performance and complexity. Overfitting occurs when the model memorizes training noise (e.g., irrelevant fluctuations) as meaningful patterns, thus reducing generalization ability. Tuning alleviates this situation by adjusting regularization parameters (lambda, gamma), limiting tree depth, or reducing ensemble size (n_estimators). Bayesian optimization (BO) effectively 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 through Bayes' theorem, which can be represented as:

[0079]

[0080] where A represents the search domain of X, P represents the objective function, X and X+ correspond to the variables to be optimized and the optimal variables, respectively. To determine the next hyperparameter combination for evaluation, BO employs an acquisition function, which is defined in equation (2) as follows:

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

[0082] where t represents the iteration number; 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 optimal solutions, while exploitation fine-tunes the search around high-performance regions; through iterative updates of the surrogate model and evaluation of new hyperparameter combinations, BO converges to the optimal hyperparameter set or terminates after the pre-defined computational budget is exhausted.

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

[0084] Before optimizing the XGBoost prediction model training, based on the specification requirements of machine learning model training, the original dataset needs to be systematically divided; the specific implementation is as follows: using random sampling technology to divide the dataset into training set, validation set and test set according to the ratio of 70%:15%:15%. Among them: the training set (70%) is used for model parameter iterative optimization; the validation set (15%) is used for hyperparameter tuning and model selection; the test set (15%) is used for final model performance evaluation.

[0085] In view of the Bayesian nature 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 physical property parameters of the integrated sand and the geometric characteristics of the pile foundation are as follows: the sand physical property parameters and the geometric parameter data of the single pile are dimensionless processed to improve the generalization ability of the trained model:

[0087]

[0088] In the above formula (3) and formula (4), D r is the relative density, is the critical friction angle, γ' is the effective unit weight, γ w is the unit weight of water, D is the pile diameter, L p is the buried depth, z is the depth, y is the lateral deformation, p is the lateral resistance of the soil body, is the dimensionless number of deformation mechanism, z / D is the depth dimensionless number, y / D is the deformation dimensionless number, is the ratio of unit weight.

[0089] S200, the six dimensionless core input variables D r , z / D, y / D and in formula (4) at the predetermined depth position of the sand are introduced into the trained XGBoost prediction model. The XGBoost prediction model will start from the root node according to the value of the input variable, traverse the branches of the decision tree in turn, and finally reach the leaf node according to the value of the input variable. Each leaf node corresponds to a predicted value, and the predicted values of all decision trees corresponding to the leaf nodes are added to obtain the final predicted value of the soil resistance p The calculation formula is as follows:

[0090]

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

[0092] Finally, a series of predicted values at different deformations on the target p-y curve are obtained

[0093] S300, a series of predicted values output by the XGBoost prediction model and the corresponding deformations are introduced into the GPR model to obtain a target smooth and continuous p-y curve.

[0094] The specific process of generating a high-precision target p-y curve is: after the predicted value output by the XGBoost prediction model and its core input variables are introduced 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 observed data:

[0095] f ~ N(μ, K) (6);

[0096] Where the covariance matrix K is defined by the kernel function k(x i ,x j ), and the RBF kernel function is:

[0097]

[0098] Reflecting the similarity between input points; then when integrating the observed data, a joint distribution is constructed to obtain the posterior distribution of the observed data (X, y) and the predicted point X * :

[0099]

[0100] Where σ is the observed noise variance, and I is the identity matrix;

[0101] Then the prior is updated using the conditional probability, that is, the prior is conditioned on the observed data y to obtain the posterior distribution of the predicted point X * :

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

[0103] The predicted value is the posterior mean μ * , and the variance Σ * provides a prediction confidence interval. The RBF kernel function is suitable for smooth p-y curves, and the kernel parameters are optimized by maximizing the marginal likelihood, thereby generating a high-precision target p-y curve.

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

[0105] 1) After completing the training of the XGBoost prediction model, the different deformations of the specified depth of the pile foundation to be evaluated and other core input feature sets are introduced into the trained and optimized XGBoost prediction model to generate a series of discrete predicted values corresponding one by one to the core input feature values.

[0106] 2) The discrete predicted values output by the XGBoost prediction model are introduced into the GPR model to generate a high-precision target p-y curve. Predicted value and the set of core input variables, i.e., the six dimensionless core input variables D in Equation (4) r 、 z / D, y / D and As input, discrete Predicted value and the set of core input variables described above as a training set to train the GPR model;

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

[0108] 4) Input the core input features for training GPR model again, and the predicted value output by GPR Connect, that is, a smooth p-y curve can be generated.

[0109] The XGBoost+GPR model in the above steps S200 and S300, Figure 1 The prediction flowchart is shown in the figure and can be divided into the following parts:

[0110] a) Dividing data set

[0111] In order to ensure accurate prediction results, the model training needs a database containing a representative sample range. Table 1 provides the database used in this study. It contains 221 p-y curve data points (a total of 2554 data points), which come from 19 representative studies on pile lateral response. Among them, 43 p-y curves are obtained from three sources through a numerically verified model that has been thoroughly verified. The data set is divided into three different subsets: training set, validation set and unseen test set. The training set and the validation set account for 70% and 15% of the total data set, respectively, and play a key role in adjusting the weight and hyperparameters of the model during the training process. The test set also accounts for 15% of the total data set, which is used to evaluate the trained model.

[0112] Table 1. Range of characteristic parameters of pile lateral response test

[0113]

[0114]

[0115]

[0116] b) eXtreme Gradient Boositng (XGBoost)

[0117] The core concept of XGBoost is to build multiple trees, where each new tree is designed to correct the mistakes of previous trees, thus 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] where is the domain of the regression tree. Here q T represents the structure of each tree that maps examples to the corresponding leaf index. T is the number of leaves on the tree. Each f k corresponds 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] Collect 2,554 p-y curve data points from 19 studies, including 7 core variables: relative density D r , critical friction angle Dimensionless number of deformation mechanism Depth dimensionless number z / D, deformation dimensionless number y / D, specific gravity ratio and dimensionless resistance Then divide into 70% training set, 15% validation set, 15% test set, and reserve 3 independent curves for generalization ability verification.

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

[0123]

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

[0125] XGBoost key hyperparameters are searched using Gaussian Process Regression (GPR) as a surrogate model; the exploration and exploitation are balanced by the acquisition function, and the optimal parameter combination is efficiently approximated; specifically, the validation set performance evaluation index CC (formula (12)) is taken as the objective function of Bayesian optimization, and the initial point is 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, for the sample data of the current node to be split, all features and their possible split points are traversed, and the feature and split point that can bring the maximum gain (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 average of the sample observed values, is the average of the sample predicted values; the termination condition is cooperatively controlled, if the validation set evaluation index has not improved for 50 consecutive rounds, stop generating new trees, and the final model is retained to the best iteration round; through the foregoing process, the XGBoost prediction model after training optimization can be generated.

[0128] c) Gaussian Process Regression (GPR)

[0129] GPR is a probabilistic and non-parametric supervised learning method, aiming to generalize the non-linear and complex function mapping hidden in the data set. By using kernel functions, GPR has been proven to be very effective in handling non-linear data. Given a training data set D = {(x i ,y i )|i=1,...,n}, the input data X ∈ R D×n is called the design matrix, and y ∈ R n is the vector representing the expected output. The main assumption of Gaussian Process Regression (GPR) is:

[0130] y = f(x) + ε (13) ;

[0131] where represents the mean square deviation of the observed output from its expected value due to measurement errors or inherent randomness in the data. In the GPR method, the n observations in the data set are considered as samples extracted from a multivariate Gaussian distribution. In Gaussian Process Regression, it is assumed that the function f(x) is 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 a 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. Choosing a suitable kernel is based on assumptions such as smoothness and possible patterns expected in the data. In this study, the main kernel function used is the Radial Basis Function (RBF) kernel combined with a white noise kernel.

[0134]

[0135] The Radial Basis Function provides a rich expressive kernel to model smooth and stationary functions. The hyperparameters θ = {l, σ f ,σ n} in the kernel function are denoted as length scale l, signal variance σ f , and noise level σ n . The length scale l describes the smoothness of the function σ f . The smaller l is, the faster the function changes, while the larger l is, the smoother the prediction is. The signal variance σ f is a scaling factor that controls the variation of the predicted values from the mean. The noise level σ n specifies the amount of expected noise in the observations (or training data).

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

[0137] Figure 2 The correspondence between the observed values and the predicted values when using the XGBoost prediction model alone is demonstrated in detail. The proposed XGBoost prediction model exhibits excellent performance metrics 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's predictions, and the Correlation Coefficient (CC) reaches 0.999, which is nearly 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 on 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 collectively confirm that the XGBoost prediction model exhibits excellent accuracy and generalization ability in evaluating the soil reaction force p at different specified depths for different soil bodies.

[0138] Figure 3Visualization of the correspondence between observed and predicted values is provided using only the Gaussian Process Regression (GPR) model. For the training set, the GPR model achieves satisfactory performance metrics: root mean square error (RMSE) of 9.160, scatter index (SI) of 0.971, and correlation coefficient (CC) of 0.329. When applied to the independent test set, the predictive performance of the model decreases. The evaluation results for the test set show that the RMSE increases to 8.830, the SI value is 1.044, and the CC decreases to 0.355. These metrics indicate that the model’s predictions for unknown data have a large deviation from the actual observations, reflecting a lack of generalization capability.

[0139] The output of the XGBoost prediction model is a set of scatter points of dimensionless lateral force These scatter points cannot be directly connected to form smooth p-y curves. The GPR model can further process the scatter point set by exploiting its ability to perform regression and smooth curves based on a limited number of data points. By combining the two models shown in the flowchart in Figure 1 The hybrid model retains the precise prediction capability of the XGBoost prediction model for soil lateral force p under specific conditions. Figure 4 The comparison between the hybrid model’s predicted values and observed values for the entire dataset is shown. The hybrid model has an RMSE of 0.445, an SI of 0.051, and a CC of 0.999. In the hybrid model, the GPR component utilizes the output values of the trained XGBoost prediction model as input, as shown in Figure 4 Therefore, the comparative analysis does not distinguish between training and test datasets. The three curves from Qi et al. (2016), Choo and Kim (2016), and Wang et al. (2021) are excluded from the training and testing process to evaluate the generalization performance of the XGBoost+GPR model. Figure 5 The comparison results in show that, although the output points of the XGBoost prediction model are tightly distributed around the corresponding p-y curves, directly connecting these points does not form a smooth curve consistent with the physical mechanism. This phenomenon occurs because it is not guaranteed that the sampling range and density of p-y curves from different sources in the training set are completely consistent. In contrast, the continuous smooth curve output by the XGBoost+GPR model closely matches the observed p-y curves.

[0140] By utilizing the XGBoost+GPR model, a series of p-y curves distributed along the pile can be obtained, which can subsequently be used to back-calculate the force-displacement and moment-depth curves of a single pile, thereby verifying the generalization capability of the prediction model. The p-y data predicted by the XGBoost+GPR model can be applied to a beam model to calculate the lateral response of the pile. As Figure 6 and Figure 7As shown, the back-calculated lateral pile responses from the predicted p-y curves agree well with the test results, indicating that the trained XGBoost+GPR model can be used to evaluate the lateral capacity of piles.

[0141] The PISA (Pile-Soil Analysis) project combined site characterization, in-situ testing and computational analysis to develop new design models for large-diameter monopiles as offshore wind turbine foundations. A total of 14 open-steel piles were installed at the Dunkirk site and subjected to lateral loading. The related studies did not involve the construction or training of predictive models. This study only compares the four typical cases mentioned in Taborda et al. (2020). Figure 8 and Figure 9 The comparison between the measured lateral responses and the predicted model responses for four different pile diameters is shown. The comparison shows that although the XGBoost+GPR model predictions of the lateral pile responses have minor discrepancies with the in-situ test results, they are closely related to the numerical models proposed by Taborda et al. (2020) or the optimized models proposed by Burd et al. (2019), respectively. These discrepancies can be due to the inherent complexity of the site geology and the unavoidable simplifications in the modeling and simulation processes. Despite these limitations, the trained predictive models have shown sufficient reliability in guiding engineering practice. In contrast to the API (2014) method, which is limited to design in homogeneous sand conditions, the proposed models provide a more flexible framework that can effectively address complex problems.

[0142] The present invention has a reasonable concept, good generalization ability, and prediction accuracy, not only providing a scientific and efficient solution for p-y curve prediction of sand monopile foundations, but also being particularly suitable for complex and variable heterogeneous 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 application, and not to limit them; although the present application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand: it can still modify the technical solutions recorded in the foregoing embodiments, or make equivalent replacement for part or all of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present application.

Claims

1. A machine learning method for predicting the Py curve of a single pile foundation in sandy soil, characterized in that, Specifically, the following steps are included: 1) First, the physical properties of the sand and the geometric characteristics of the pile foundation are integrated as core input variables to train and optimize the XGBoost prediction model; specifically: Dimensionless processing is performed on the physical properties of sand and the geometric parameters of a single pile: (3)； (4)； In equations (3) and (4) above, D r It is relative density. It is the critical friction angle, γ' is the effective specific weight, γ w Let D be the unit weight of water, D be the pile diameter, and L be the weight of the pile. p Let z be the burial depth, y be the lateral deformation, and p be the lateral resistance of the soil. Let z / D be the dimensionless number of the deformation mechanism, z / D be the dimensionless number of depth, and y / D be the dimensionless number of deformation. The proportion of severe cases; 2) Subsequently, a series of soil reaction forces p and their related core input variables on the py curve at a specified depth location of the target single pile are input into the trained XGBoost prediction model to obtain accurate predicted values ​​of p, i.e.: The six dimensionless core input variables D at a predetermined specific depth of sand, i.e., in formula (4) r , , z / D, y / D and Imported into the trained XGBoost model, the XGBoost model, based on the previously trained decision tree structure, starts from the root node and traverses the branches of the decision tree sequentially according to the values ​​of the input variables, eventually reaching the leaf nodes. Each leaf node corresponds to a predicted value. The predicted values ​​of all leaf nodes are added together to obtain the final predicted value of the soil reaction force p. The calculation formula is as follows: （5）； in This indicates that the k-th decision tree is a pair of inputs. The predicted value; Finally, we obtained the predicted values ​​of the target py curve under a series of different deformations. ; 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; the construction process of the GPR model is as follows: 3.1) Given a training dataset Input data It is called the design matrix, and The vector representing the expected output; The different deformations of the pile foundation at a specified depth to be evaluated, along with other core input feature sets, are imported into the trained and optimized XGBoost prediction model to generate a series of discrete values ​​that correspond one-to-one with the core input feature values. Predicted value; 3.2) Discrete outputs of the XGBoost prediction model The predicted value and the set of core input variables are the six dimensionless core input variables D in formula (4). r , , z / D, y / D and As input, discrete The predicted values ​​and the set of the aforementioned core input variables are used as the training set to train the GPR model. 3.3) Constructing a Gaussian process using the RBF kernel function The RBF kernel function is suitable for smooth Py curves, and optimizes the kernel parameters by maximizing the marginal likelihood. 3.4) Re-input the core input features used to train the GPR model, and output the predicted values ​​of the GPR model. Connecting the lines will generate a smooth Python curve.

2. The machine learning prediction method for the Py curve of a single pile foundation in sand as described in claim 1, characterized in that, Before training and optimizing the XGBoost prediction model in step 1), the original dataset must be systematically divided according to the specifications for machine learning model training. Specifically, the dataset is divided into a training set, a validation set, and a test set in a ratio of 70%:15%:15% using random sampling techniques. 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 machine learning prediction method for the Py curve of a single pile foundation in sandy soil as described in claim 1, characterized in that, The specific process of training and optimizing the XGBoost prediction model in step 1) is as follows: Starting with a set of initial hyperparameters, the objective function is modeled as a Gaussian process through Bayesian optimization, where each evaluated combination of hyperparameters updates the model using Bayes' theorem. This process is represented as: （1）； Where A represents the search domain of X, P represents the objective function, and X and X+ correspond to the variable to be optimized and the optimal variable, respectively; to determine the next combination of hyperparameters for evaluation, Bayesian optimization uses a sampling function I(X), and the sampling function I(X) is defined by the following equation (2): （2）； Where t represents the number of iterations; the acquisition function balances two conflicting objectives: exploration, which involves testing unknown regions of the hyperparameter space; and development, which focuses on refining promising regions identified by the current model. Exploration ensures that the algorithm avoids getting trapped in local optima and uses exploration to fine-tune the search around high-performance regions. By iteratively updating the surrogate model and evaluating new combinations of hyperparameters, Bayesian optimization converges to the optimal set of hyperparameters or terminates when the predefined computational budget is exhausted.

4. The machine learning prediction method for the Py curve of a single pile foundation in sandy soil as described in claim 1, characterized in that... The specific process of generating the high-precision target py curve in step 3) is as follows: After importing the predicted values ​​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. （6）； The covariance matrix K is determined by the kernel function k(x) i ,x j The RBF kernel function is defined as follows: （7）； This reflects the similarity between input points; subsequently, when integrating the observation data, a joint distribution is constructed, linking the observation data (X, y) with the point to be predicted X. * The posterior distribution: （8）； in, The variance of the observed noise is represented by I, which is the identity matrix. Then, the prior is updated using conditional probability, that is, the prior is conditionalized using the observed data y to obtain the predicted point X. * The posterior distribution: （9）； The predicted value is the posterior mean μ * Variance Σ * Provides prediction confidence intervals. The RBF kernel function is suitable for smooth Py curves. It optimizes the kernel parameters by maximizing the marginal likelihood, thereby generating a high-precision target Py curve.

5. The machine learning prediction method for the Py curve of a single pile foundation in sandy soil as described in claim 1, characterized in that, 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: (10)； in, It is the domain of the regression tree; q T This represents the structure of each tree that maps the examples to the 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 w at each leaf. i Let represent the fraction on the i-th leaf; Collect data points for the Python curve, including 7 core variables: relative density D. r Critical friction angle Deformation mechanism dimensionless number Dimensionless depth z / D, dimensionless deformation y / D, and density ratio and dimensionless resistance Then, the dataset was divided into 70% training set, 15% validation set, and 15% test set, and three independent curves were retained for generalization capability verification. 1.2) XGBoost Prediction Objective Function Design The objective function L consists of the loss function l and the regularization term Ω: (11)； In equation (11) above, 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 tradeoff, maintaining simplicity and prediction accuracy to prevent overfitting; 1.3) Hyperparameter Optimization Gaussian process regression (GPR) is used as a surrogate model to search for key hyperparameters of XGBoost; by balancing exploration and utilization through acquisition functions, the optimal parameter combination is efficiently approximated. 1.4) Tree Model Training and Optimization In XGBoost, the exact greedy algorithm is used for feature splitting of decision tree nodes. Specifically, when constructing a single decision tree, for the sample data of the node to be split, it traverses all features and their possible split points, and selects the features and split points that bring the maximum gain for splitting. (12)； Where m is the number of data samples. Let be the predicted value for the i-th sample. For the observation value of the i-th sample, The average of the sample observations. The average value of the sample predictions; the termination condition is a collaborative control: if the evaluation index of the validation set does not improve after 50 consecutive rounds, the generation of new trees is stopped, and the final model is retained until the optimal iteration round; The trained and optimized XGBoost model can be generated by following the steps 1.1)-1.4) above.

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