Analysis of Offshore Pile–Soil Interaction Using Artificial Neural Network

Abstract

Offshore wind power is one of the primary forms of utilizing marine green energy in China. Currently, near-shore wind power predominantly employs monopile foundations, with designs typically being overly conservative, resulting in high construction costs. Precise characterization of the interaction mechanisms between marine piles and surrounding soils is crucial for foundation design optimization. Traditional p-y curve methods, with simplified fitting functions, inadequately capture the complex pile–soil behaviors, limiting predictive accuracy and model uncertainty quantification. To address these challenges, this research collected 1852 empirical datasets of offshore wind monopile foundation pile–soil interactions, developing p-y curve and horizontal displacement prediction models using artificial neural network (ANN) expressions and comprehensive uncertainty statistical analysis. The constructed ANN model demonstrates a simple structure with satisfactory predictive performance, achieving average error margins below 6% and low to moderate prediction accuracy dispersion (26%~45%). In contrast, traditional p-y curve models show 30%~50% average biases with substantial accuracy dispersion near 80%, while conventional finite element methods exhibit approximately 40% error and dispersion. By strictly characterizing the probability cumulative function of the neural network model factors, a foundation is provided for reliability-based design. Through comprehensive case verification, it is demonstrated that the ANN-based model has significant advantages in terms of computational accuracy and efficiency in the design of offshore wind power foundations.

1. Introduction

Although fossil fuels such as coal and oil remain the backbone of the global energy mix, offshore wind power—a renewable and low-carbon energy source—has emerged as a critical component of international efforts to achieve carbon neutrality and energy transition [1]. Rapid advancements in technology, coupled with cross-border policy coordination and collaboration (e.g., in the North Sea [2,3], Baltic Sea [4], and the Guangdong-Hong Kong-Macao Greater Bay Area [5] as a regional exemplar), have propelled offshore wind into a vital supplement to the world energy supply. In near-shore shallow water areas, offshore wind turbine foundations primarily utilize large-diameter steel pipe piles. Statistics indicate that approximately 75% of marine engineering structures currently employ large-diameter monopile foundations [6,7].

The ideal objective in pile foundation design is to balance safety with economic efficiency. However, due to the potential catastrophic risks associated with complex wind, wave, and current loads in marine environments [8], coupled with nonlinear pile–soil interactions and the significant uncertainty in existing pile design methodologies [9,10], designers are compelled to adopt conservative approaches to ensure the stability and service life of offshore platform structures while hedging against the exorbitant repair costs of pile failure [11]. The direct consequence of overly conservative design is elevated construction costs. Accurately quantifying pile–soil interaction is the key to designing marine pile foundations that balance safety with economic considerations.

Currently, common methods for analyzing pile–soil interaction primarily include theoretical approaches [12,13], empirical methods [14,15], and numerical techniques [16,17,18]. Theoretical methods typically simplify piles as idealized models such as cantilever beams [19,20,21]; however, due to excessive simplification, these approaches fail to adequately account for complex factors in pile–soil interaction, including soil nonlinear mechanical properties and variations in pile–soil interface friction [22]. Empirical methods are analytical approaches developed from extensive engineering practice or experimental data, with the p-y curve method being a typical representative model. Due to its simple expression and convenient application, this method has been widely adopted in various international specifications [23,24,25]. The method constructs empirical expressions between pile displacement and soil reaction force by fitting experimental data [26,27].

The accuracy of the p-y curve method depends on the quantity of fitted data and the functional expressions employed. Additionally, the p-y curve method exhibits significant regional characteristics and typically cannot be directly applied to different regions [28,29]. Given the extreme complexity of marine environments, pile–soil interactions exhibit significant variability and uncertainty [30,31], imposing substantial limitations on applying the p-y curve method. Empirical investigations have demonstrated that the p-y curve method exhibits significantly compromised predictive accuracy when applied to large-diameter pile foundations and under specialized geotechnical conditions [32,33,34,35].

The finite element method (FEM) is a widely utilized numerical technique capable of accurately characterizing the boundary and initial conditions of pile–soil interaction systems [36]. However, the highly nonlinear mechanical properties inherent in marine soils present significant challenges to predictive modeling. Although FEM simulations are generally unbiased on average, they exhibit moderate variability in predictive accuracy due to these nonlinearities [37,38]. When confronted with complex environmental loads and unique geological formations, the FEM not only struggles to maintain precision but also encounters substantial hurdles, including significantly increased computational costs and prolonged processing times [39,40]. These limitations pose serious obstacles in practical engineering applications that demand high computational efficiency and rapid iterative analyses.

In recent years, machine learning methods have been widely applied to complex geotechnical engineering problem analysis due to their powerful nonlinear representation capabilities [41,42,43,44]. For example, support vector machines (SVMs) demonstrate high accuracy in landslide susceptibility analysis and rock deformation prediction [45,46], while artificial neural networks (ANNs) excel in soil classification [47] and in predicting earth dam settlement and foundation settlement [48]. Machine learning methods effectively compensate for the deficiencies of traditional techniques in handling highly nonlinear problems, overcoming the oversimplification of theoretical approaches, the applicability limitations of empirical methods, and the accuracy and computational cost issues of numerical methods, making them highly suitable for analyzing pile–soil interactions in specialized marine geotechnical environments.

This study compiles a comprehensive dataset encompassing multi-dimensional parameters of marine large-diameter monopile foundations, including marine geological conditions, pile design specifications, horizontal displacement profiles, and soil pressure distribution characteristics. Leveraging this dataset, we develop a neural network model specifically tailored for analyzing pile–soil interactions in marine environments. The proposed framework achieves enhanced predictive accuracy for pile–soil interaction behaviors through systematic model training and rigorous application while significantly improving computational efficiency. This methodology substantially mitigates design risks associated with marine large-diameter pile foundations and provides a robust technical framework for cost-effective engineering solutions in offshore pile construction.

The subsequent section describes the database development process and its contents, while Section 3 details the construction of the neural network model, including fundamental concepts, development procedures, and the selection of key parameters. Section 4 and Section 5 analyze the prediction results of the neural network model, discuss its performance and influencing factors, and validate the model’s reliability using a real-world case study. Finally, the paper concludes with a summary of the research content and findings.

2. Database

This study collected 1852 datasets comprising field measurements and model test data from 29 monopile foundations across 24 offshore wind projects worldwide. Leveraging the compiled datasets, we established a marine geotechnical database to systematically investigate and predict pile–soil interactions of monopile foundations under complex loading and geological environments.

The database systematically documents three critical components for each monopile (

Table 1. Soil information and pile design parameters of the pile foundation in the database.

Pipe Number	Soil Type	Load	Soil Parameter				Pile Parameter			Reference
		F (kN)	φ (°)	c (kPa)	E (MPa)	γ (kN/m3)	L (m)	D (m)	t (mm)
P1	Clay, fine sand	200–650	18–42	0–15	10–180	18.5–20.5	78.5	2	30	[49]
P2	Clay, fine sand	200–650	18–42	0–15	10–180	18.5–20.5	78.5	2	30	[49]
P3	Medium-coarse sand, residual cohesive soil	148–1480	14–35	23–24	15–30	23.5–27.5	55	1.8	30	[50]
P4	Medium-coarse sand, residual cohesive soil	148–1480	14–35	23–24	15–30	23.5–27.5	53	1.9	30	[50]
P5	Silt	140–840	3.2–34.0	2–37	69–157	18.0–19.5	51	1.8	25	[51]
P6	Silty clay	400–800	8.1–33.6	7–37	8.0–33.6	17.8–20.0	72.7	2	-	[52]
P7	Marine clay, silty clay, residual soil	200–900	24–33	-	-	17.5–20.5	26.6	1.016	16	[53]
P8	Silt, fine sand, silty clay	150–900	31–35	-	-	6.5–9.5	89	2	26–30	[54]
P9	Silt, fine sand, silty clay	40–480	11–34.5	0.1–16.6	5.9–36.7	17.4–19.5	85.2	1.7	25–30	[55]
P10	Silt, silty clay, clay	40–480	11–34.5	0.1–16.6	5.9–36.7	17.4–19.5	85.2	1.7	25–30	[55]
P11	Silt, silty clay, medium sand, silty sand	100–700	30–37	-	2–50	5.8–10.9	105.4	2.4	40	[56]
P12	Clay	300–1300	-	-	16.02	17.9	83	2	26	[57]
P13	Soft clay, sandy	200–2000	35	-	12–17	6.7	66	2.2	30	[58]
P14	Silt, sandy silt, silty clay, fine sand	22–220	15–34	11–54.7	-	17.3–19.8	60	1.2	16	[59]
P15	Silty clay, clay, silt	20–300	8–34	7–18	8–36	17.4–20.5	82.1	30	1.7	[60]
P16	Silty clay, fine sand	50–425	8–35	2–17	20–60	18.3–20.0	93.7	35	2.8	[61]
P17	Silty clay, fine sand	100–500	8–35	2–17	20–60	18.3–20.0	93.7	35	2.8	[61]
P18	Silt, silty, sand	300–2000	8–18	15–16	8–30	17.7–20.2	70.0	30	2.2	[62]
P19	Silt, argillaceous sand	50–250	8–45	1–20	8–180	23.4–27.5	30.7	16	1.4	[63]
P20	Silt, argillaceous sand	50–250	8–45	1–20	8–180	23.4–27.5	32.2	16	1.4	[63]
P21	Sand	250–1500	38–43	-	-	8.1–19.5	0.41	0.079	1.2	[64]
P22	Silty soil	0–10	25	12	-	-	2.3·	0.089	4.0	[65]
P23	Silty soil	8.1–55.1	27	5	-	-	1.1	0.032	7.0	[66]
P24	Sand	0–3.8	38	-	8.23	15.1–16.5	1.4	0.102	6.4	[67]
P25	Sand	0.28–2.6	28.5	-	-	17.5	7.0	0.114	2.5	[68]
P26	Silt	0.26–1.5	35.5	0.1	-	15.7	2.0	0.165	3.0	[69]
P27	Silt, sand	0.72–6.1	30	-	-	19.3	4.5	0.159	4.5	[70]
P28	Sand	400–800	37	30	2.0	-	3.0	0.34	-	[71]
P29	Sand	148–1480	-	30	1.8	-	12	2.0	-	[72]

): (i) soil parameters encompassing stratigraphic classification and strength characteristics (φ: internal friction angle; c: cohesion; E: elastic modulus; γ: soil’s unit weight, et al.); (ii) geometric design specifications of steel tubular piles (L: pile length; D: outer diameter; t: wall thickness, et al.); and (iii) operational loading regimes with horizontal force (F) applied at the pile head. Depth-dependent soil–structure interaction data are parametrized by l (depth from pile head), with pile geometric configuration schematically detailed in Figure 1.

Field-monitored monopiles were predominantly embedded in clay and sand strata, with three cases (P3, P4, and P19) terminating in weathered rock formations. Laboratory tests employed reconstituted sand-silt mixtures. The soil unit weight (γ) exhibited limited variation, predominantly between 18.8 and 19.5 kN/m³. While strength parameters demonstrated significant heterogeneity across the dataset. As illustrated in Figure 2, internal friction angle (φ) varied considerably from 5° to 42°; cohesion (c) predominantly clustered within 11–22 kPa, with only approximately 5% of specimens exceeding 50 kPa; the distribution of shear strength (c_u) exhibited considerable variability, ranging from 10 kPa to 80 kPa, with values predominantly concentrated within the 35–42 kPa interval; and soil elastic modulus (E) exhibited a wide distribution from 6 to 180 MPa, with half below 30 MPa and 80% below 69 MPa.

Figure 3 delineates the nonlinear correlation between lateral soil displacement (y) and mobilized soil resistance (p). Soil pressure on piles in clay typically ranged from 55 to 120 kN/m, whereas sand deposits demonstrated enhanced lateral resistance (215–412 kN/m) attributable to particulate interlocking and dilatant behavior.

Field tests revealed significant discrepancies in horizontal displacement magnitudes between clay- and sand-embedded piles. The clay-pile systems exhibited maximum displacements up to 255 mm, contrasting sharply with the sand systems’ 122 mm maximum. The model test exhibited considerable scatter, with maximum displacements reaching 500 mm; however, 80% of the values were below 35 mm. This divergence underscores limitations in model testing: while capturing general load-displacement trends, scaled tests inadequately replicate in situ stress redistribution processes and strain localization patterns, particularly under large deformation regimes.

The database reveals geometric scaling characteristics between prototype and model piles. Prototype monopile lengths (L) generally exceeded 50 m, with maximum values reaching 105.4 m, whereas model piles exhibited significantly shorter dimensions ranging from 0.41 to 12.0 m. Diameter (D) distributions show marked divergence: prototype piles cluster in the 1.0–2.4 m range (80% within 1.8–2 m), while model diameters span three orders of magnitude (D: 0.032–2 m). Slenderness ratios (L/D) predominantly occupy 20–50 across both groups, satisfying prototype and model piles geometric proportional control criteria [73].

As illustrated in Figure 4a, normalized horizontal displacement (y/L) exhibits inverse proportionality to relative depth (l/L) in both clay and sand. Thickness (t) of field-measured piles ranged from 16 to 30 mm, with a predominance of 30 mm, while model piles featured substantially thinner walls of only 1.2–7 mm. Notably, geometric parameter covariation emerges: increasing L correlates with proportional D and t enhancements, reflecting design optimization for structural stability and bearing capacity across varying embedment depths.

To satisfy the installation requirements of steel pipe piles and prevent local buckling when the pile reaches its yield strength, the thickness needs to comply with the following criterion [23]:

$$t\ge 6.35+\frac{D}{100}$$

(1)

Figure 4b illustrates the relationship between thickness and diameter for piles in the database, compared with data from other case studies [74]. The straight line in the figure represents the minimum thickness value recommended by the API specification. A comparative analysis reveals that most field-tested piles in the database adhered to this standard, whereas the model test piles exhibited only basic compliance.

3. Neural Network Modeling of Pile–Soil Interaction

3.1. Artificial Neural Networks

Artificial neural networks (ANNs) are biologically inspired computational frameworks designed to emulate the information processing mechanisms of biological neural systems. As depicted in Figure 5, a standard ANN architecture consists of three hierarchical layers:

(i). Input Layer: Receives normalized geotechnical parameters (e.g., pile diameter D, soil elastic modulus E).

(ii). Hidden Layer: Extracts nonlinear relationships through weighted transformations and activation functions.

(iii). Predicts target variables (e.g., pile head displacement, soil reaction forces).

The performance of the ANN depends on the number of layers and neurons. Increasing neurons or adjusting weights reduces the mean squared error (MSE).

3.2. Architecture Design & Workflow

3.2.1. Data Preprocessing

Input–output parameters are normalized to the interval [0, 1] to mitigate numerical instability caused by disparate units (e.g., MPa vs. mm):

$$x_{norm}=\frac{x-x_{min}}{x_{max}-x_{min}}$$

(2)

where $x_{norm}$ is the normalized value; x is the raw data, and $x_{max}$ or $x_{min}$ denote the maximum or minimum values in the dataset.

3.2.2. Topology Structure

The hidden layer dimensionality is determined via Kolmogorov’s superposition theorem [75], which prescribes a theoretical upper bound for the number of hidden nodes (m) as:

$$m=2n+1$$

(3)

where n corresponds to the input dimension. This configuration balances model complexity and generalization capacity.

Equation (3) establishes the theoretical upper bound for nodes in the hidden layer. To systematically evaluate optimal network architecture, we conducted a parametric study by training networks with incrementally increasing hidden layer complexity (3, 5, 7, 9, 11, and 13 nodes) and assessed their performance against established metrics. Figure 6 plots the coefficient of determination (R²) across training, validation, and test datasets. Increasing node count initially enhances model performance, as evidenced by rising R² values. However, beyond 5 nodes, R² exhibits pronounced fluctuations across all datasets. Networks with more than 9 nodes show a declining R² trend in training and test datasets, indicative of overfitting. This suggests that in geotechnical modeling, a moderate number of nodes (5–7 nodes) optimizes generalization capability, while excessive complexity degrades validation/test accuracy.

3.2.3. Propagation Mechanism & Training Algorithm

For an input vector x = [x₁, x₂,…, x_n]^T, the pre-activation (z_k) and activation (a_k) of the k-th hidden neuron are computed as:

$$z_k=\sum _{p=1}^s{n}_{k,p}{w}_{k,p,t}+b_{k,t}$$

(4)

$$a_k=f\left(z_k\right)$$

(5)

where w_k,p,t and b_k,t denote the weight matrix and bias vector of the layer, respectively. f(x) represents activation functions (logistic) used to convey interlayer information.

The Levenberg–Marquardt (LM) optimization algorithm is employed for network training due to its hybrid mechanism that synergistically integrates the robustness of gradient descent in shallow regions of the error surface with the quadratic convergence properties of the Gauss–Newton method near minima. The weight update rule is formulated as:

$$\mathsf{\Delta }W=-{\left(J^{T}J+\mu I\right)}^{-1}{J}^{T}e$$

(6)

where $J$ is the Jacobian matrix of partial derivatives, μ is a damping factor, and e is the residual vector.

3.2.4. Learning Rate & Iterations

The learning rate (η) regulates the step magnitude in gradient-based optimization, critically balancing convergence stability and rate. Elevated η values (e.g., η > 0.1) accelerate convergence yet risk overshooting minima or inducing oscillatory behavior, whereas diminished values (η < 0.001) prioritize stability at the cost of computational inefficiency. This study adopts η = 0.01, a heuristically calibrated value within the empirically effective range η ∈ [0.001, 0.1] for geotechnical systems [76,77,78].

Concurrently, the maximum iteration threshold (N_max = 1000) safeguards against overfitting by constraining parametric updates, with training termination triggered upon either reaching N_max or validation loss plateau ($\Delta \mathcal{L}$ < 10⁻⁵ sustained over 50 epochs).

3.2.5. Validation and Regularization

To prevent overfitting, a 70-15-15 data partitioning (training-validation-testing) is implemented, coupled with Bayesian regularization to penalize excessive weight magnitudes:

$$L_{reg}=\beta L+\alpha \sum _i{\omega }_i^2$$

(7)

where α and β are hyperparameters optimized via cross-validation.

4. Results

4.1. Predictions of p-y Curves

A critical consideration lies in the selection of input parameters for the neural network model. While using all parameters from the database as inputs may theoretically maximize data utilization, it introduces computational complexity and risks overfitting due to redundant or low-sensitivity variables. Thus, the selection of which influencing factors to incorporate as input parameters is a task that requires careful engineering judgment. Fortunately, based on the physico-mechanical properties of clay and sand foundations and in conjunction with prior research [79,80,81], the primary influencing factors to be used as input parameters can be identified.

For the clay model, the selected inputs were pile diameter (D), length (L), depth (d), ultimate resistance (p_u), undrained shear strength (c_u), and lateral displacement (y). Similarly, the sand model incorporates D, L, d, internal friction angle (φ), soil unit weight (γ), and y as inputs, with soil pressure (p) as the output. The architecture of the clay p-y curve neural network is illustrated in Figure 7, while detailed structural hyperparameters for both models are tabulated in

Table 2. Neural network model structure parameters and predictive regression values.

Prediction	Input Nodes	Output Node	Hidden Nodes Number	Learning Rate	Maximum Iteration	R2
						Train	Validation	Test
Clay p-y curve	D, L, d, pu, cu, y (6)	p	5	0.01	1000	0.95	0.96	0.97
Sand p-y curve	D, L, d, φ, γ, y (6)	p	4			0.93	0.96	0.96
Pile horizontal displacement	F, L, D, d, l, E, φ, c (8)	y	5			0.97	0.96	0.98

.

It is also crucial to highlight another issue: the potential existence of size effects means that the results of model tests cannot fully replace those of field tests. Therefore, the data from model tests and field tests need to be separately employed to establish predictive models (primarily focused on sand). For ease of description in subsequent discussions, the predictive results from model tests and field tests are combined, and the use of ratios can effectively characterize these predictive outcomes. This will be elaborated in later sections.

The hidden layer neuron outputs are computed via the activation function as:

$$N^1=1-\frac{2}{e^{\left[2\left(W_{01}{N}^0+B_{01}\right)\right]}+1}$$

(8)

where W₀₁ (5 × 6 weight matrix) and B₀₁ (5 × 1 bias vector) govern the input-to-hidden layer transformations.

The output layer is expressed as:

$${\widehat{Y}}_{p,k}=Y_{(m,min)}+\left\{1-\frac{2}{\mathit{exp}\left[2\left(W_{12}{N}^1+B_{12}\right)\right]+1}\right\}\left(Y_{(m,max)}-Y_{(m,min)}\right)$$

(9)

where $Y_{m,min}$ and $Y_{m,max}$ denote the normalized bounds of measured input parameters (e.g., D, L, d, p_u, c_u, φ, γ, y) with subscript m indicating experimental measurements. W₁₂ and B₁₂ represent the hidden-to-output layer weight matrix and bias vector, respectively.

The squared error (${\epsilon }_k^2$) is calculated by comparing the observed values with their corresponding predicted values generated by the model.

$${\epsilon }_k^2={\left({\widehat{Y}}_{p,k}-Y_{m,k}\right)}^2$$

(10)

By iterating over all samples in the dataset and aggregating these squared differences, the mean squared error (MSE) is obtained as a quantitative measure of the model’s performance.

$${\epsilon }^2={\sum }_{k=1}^i{\epsilon }_k^2/i$$

(11)

The MSE serves as the objective function in the training of the artificial neural network. Minimizing the MSE through appropriate optimization algorithms allows for the determination of the optimal weights and biases (W₀₁, W₁₂, B₀₁, B₁₂).

The optimized parameters of the neural network for clay p-y curves are mathematically expressed as follows:

$$\begin{array}{ll}& w_{01}=\left[\begin{array}{cccccc}-9.45& -6.45& -1.90& -4.33& -4.38& -0.09\\ -0.79& 0.49& -0.41& 1.87& -1.76& 5.65\\ -6.83& 7.93& -12.76& 7.72& -13.25& 3.78\\ -2.92& 3.42& -3.98& 1.39& -9.06& 0.89\\ 2.22& 0.08& 2.29& -3.61& 0.54& 0.24\end{array}\right]\\ & w_{12}=\left[\begin{array}{lllll}1.54& 2.18& 1.01& -3.05& 2.73\end{array}\right]\end{array}$$

(12)

$$B_{01}=\left[\begin{array}{c}4.0207\\ 5.8339\\ 0.1103\\ -2.735\\ -3.251\end{array}\right],B_{12}=\left[-0.027\right]$$

(13)

Taking this model as an example, the simplified expression is as follows:

$$N^1=\left[\begin{array}{c}{n}_1^1\\ n_2^1\\ n_3^1\\ n_4^1\\ n_5^1\end{array}\right]=\left[\begin{array}{c}f\left(-6.828D-0.082L-0.142d-0.008p_u-0.137c_u-0.0004y+14.972\right)\\ f\left(-0.573D+0.006L-0.031d+0.003p_u-0.055c_u+0.023y+6.59\right)\\ f\left(-4.935D+0.101L-0.954d+0.014p_u-0.414c_u+0.015y+5.754\right)\\ f\left(-2.109D+0.043L-0.298d+0.002p_u-0.283c_u+0.004y+0.933\right)\\ f\left(1.603D+0.001L+0.172d-0.006p_u+0.017c_u+0.001y-4.687\right)\end{array}\right]$$

(14)

$$\begin{array}{ll}{\widehat{Y}}_{p.k}& =\left[p_p\right]\\ & =\left[1.186+\left(1-\frac{2}{e^{2\left(1.5442n_1^1+2.175n_2^1+1.0083n_3^1-3.0477n_4^1+2.7265n_5^1-0.027\right)}+1}\times 577.443\right]\right.\end{array}$$

(15)

where subscript p denotes predicted values from the neural network, contrasting with subscript m for measured data. f(x) represents the activation function.

For sand soils, the optimal weights and biases are derived as:

$$\begin{array}{ll}& w_{01}=\left[\begin{array}{cccccc}-0.736& 0.744& -1.277& -0.114& 0.543& -1.756\\ -0.469& -1.337& 1.254& 0.092& 0.574& 1.524\\ -2.774& 1.539& -2.097& -0.011& 1.499& -2.119\\ 0.777& -1.179& -1.835& 0.729& 0.485& -0.239\end{array}\right]\\ & w_{12}=\left[\begin{array}{llll}2.6457& 3.3194& -2.3985& -2.2238\end{array}\right]\end{array}$$

(16)

$$B_{01}=\left[\begin{array}{c}-0.296\\ 1.4104\\ -0.098\\ 1.6015\end{array}\right],B_{12}=\left[-1.1929\right]$$

(17)

As illustrated in Figure 8, the neural network demonstrates outstanding predictive performance. For the clay p-y curve model, both the training and testing phases exhibited strong convergence (R² ≥ 0.95). The neural network model for sand p-y curves demonstrated robust correlation with experimental data, achieving a test set R² = 0.96. This statistical validated confirms the model’s high predictive accuracy and reliability for both sandy soils and clayey soils.

4.2. Pile Horizontal Displacement Prediction

The data were randomly extracted from the pile horizontal displacement database to construct an ANN-based model for predicting pile horizontal displacement. The structural parameters of the model are presented in Table 2. The input parameters include pile horizontal load (F), pile length (L), pile diameter (D), penetration depth (d), distance from the test point to the pile top (l), soil elastic modulus (E), internal friction angle (φ), and cohesion (c), with the pile horizontal displacement serving as the output parameter. Given the strong correlation between t and D as well as L, and considering that the research focus lies on the global horizontal displacement of the pile rather than its internal deformation characteristics, the input parameters of the model excluded the pile t.

For the constructed ANN model of pile horizontal displacement, the optimal values of parameters W₀₁, W₁₂, and B₀₁, B₁₂ are determined.

$$\begin{array}{ll}& w_{01}=\left[\begin{array}{cccccccc}-1.518& 2.242& 0.759& -3.152& 1.626& 0.665& -1.335& -1.246\\ 2.355& -2.595& -1.125& 2.166& 0.388& -0.646& -0.078& 0.094\\ 0.398& 0.951& -0.331& -0.796& -0.493& 1.546& 0.562& -0.0402\\ 0.299& 1.628& 3.163& 0.779& 0.238& 2.878& -4.888& -1.2601\\ 3.443& 1.513& 1.561& -0.172& -0.627& 3.138& 2.378& -0.342\end{array}\right]\\ & w_{12}=\left[\begin{array}{lllll}-1.9394& 1.6094& 2.5132& 1.3983& 3.1218\end{array}\right]\end{array}$$

(18)

$$B_{01}=\left[\begin{array}{c}-1.2761\\ 1.8069\\ 1.1858\\ 3.4579\\ -3.103\end{array}\right],B_{12}=\left[-0.91565\right]$$

(19)

The training and testing performance metrics of the model are systematically presented in Table 2 and Figure 8. The ANN framework demonstrated robust convergence characteristics, achieving exceptional agreement between predicted and observed displacements in the training dataset (R² =0.97). During validation and testing phases, the model maintained superior predictive capability with R² > 0.96 for both phases, confirming its reliability in horizontal displacement prediction across diverse geotechnical scenarios.

5. Discussion

5.1. Model Performance

To rigorously evaluate the performance of the neural network model, the model factor (λ) is used to evaluate the accuracy of the model:

$${\lambda }_p=\raisebox{1ex}{$p_m$}\!\left/ \!\raisebox{-1ex}{$p_p$}\right.$$

(20)

$${\lambda }_y=\raisebox{1ex}{$y_m$}\!\left/ \!\raisebox{-1ex}{$y_p$}\right.$$

(21)

where P_m and y_m denote measured soil resistance and displacement, respectively, while P_p and y_p represent predicted soil resistance and displacement, respectively.

Figure 9 presents the predicted p-y curves obtained from the neural network model and the API specification. It can be observed that the predictions from the neural network model closely align with the measured values along the 1:1 diagonal line, demonstrating a high degree of consistency. Only when the soil resistance is relatively low (less than 50 kN/m) do the data points exhibit some dispersion. In contrast, the values calculated using the API specification show significant deviations from the actual measurements, underestimating clay resistance by approximately 50% and overestimating sand resistance by about 30%, with a greater degree of data scatter.

For the clay p-y curves, the model factor λ of the neural network model has a mean value ${\mu }_{\lambda p}$ = 1.02 and a coefficient of variation COV_λ = 0.45. Compared to the clay p-y curves predicted by the API specification (μ_λ = 1.51, COV_λ = 0.73), the mean model factor of the neural network model is closer to 1. Similarly, for the sand p-y curves, the neural network model yields a mean model factor μ_λ = 1.06 with a coefficient of variation COV_λ = 0.29. In comparison, the API specification predicts a mean model factor of μ_λ =0.70 and COV_λ =0.79 for sand p-y curves.

These results demonstrate that the neural network model provides more accurate predictions, with average errors ranging from only 2% to 6%. According to the model accuracy classification [82], the variability in the prediction accuracy of the neural network model is classified as moderate to low. Overall, the soil resistance values computed by the neural network model, utilizing the selected input parameters and hidden nodes, effectively reflect the actual soil resistance behavior.

The measured data from field tests on the horizontal bearing capacity of offshore wind turbine pile foundations were input into the trained neural network model to obtain predicted values of horizontal displacement. These predicted values were then compared with the actual measured horizontal displacements, as illustrated in Figure 10. Observing the distribution of the data points, it is evident that the predictions from the neural network model are predominantly concentrated near the line along the 1:1 reference line, indicating low data dispersion. Specifically, the mean value of the model factor is μ_λ = 0.9, with a coefficient of variation COV_λ = 0.26.

To achieve a more rigorous comparative analysis, a finite element model (FEM) was utilized to predict the horizontal displacement of offshore wind turbine monopile foundations. For conciseness, the detailed description of the FEM is omitted here; the specific methodologies and parameters can be found in the work of [38]. The FEM results exhibited considerable dispersion and lower predictive accuracy, significantly underestimating the horizontal displacement around the pile. Specifically, the average value of the μ_λ was 1.41 with COV_λ = 0.4. The deviation of this result is closely related to the establishment of the model and the selection of data, and it may be attributed to issues such as the mixed use of model data and on-site measured data. This clearly demonstrates that the neural network model offers significant advantages and higher accuracy in predicting the horizontal displacement of offshore monopile foundations.

5.2. Parameter Sensitivity

Table 3. Spearman’s rank correlation test results.

Model	Parameters	Spearman’s Rank
		p	ρ
Clay p-y curve	(λ, Pu)	0.039	0.142
	(λ, d/L)	0.023	0.183
	(λ, y/D)	0.028	0.153
	(λ, y/L)	0.150	0.072
	(λ, D/L)	0.048	0.109
	(λ, D)	0.046	−0.106
	(λ, L)	0.037	−0.125
Sand p-y curve	(λ, φ)	0.031	−0.163
	(λ, d/L)	0.046	−0.147
	(λ, y/D)	0.029	0.191
	(λ, y/L)	0.023	−0.178
	(λ, t/D)	0.068	−0.032
	(λ, D/L)	0.010	0.201
	(λ, D)	0.000	−0.401
	(λ, L)	0.000	−0.272
Pile horizontal displacement	(λ, φ)	0.000	−0.191
	(λ, F)	0.000	0.219
	(λ, E)	0.000	−0.241
	(λ, l/L)	0.001	0.106
	(λ, t/D)	0.001	−0.111
	(λ, D/L)	0.003	−0.093
	(λ, D)	0.000	0.158
	(λ, L)	0.000	−0.338

presents the Spearman rank correlation coefficients between the model factor λ and each input parameter at a significance level of 0.05. For the clay p-y curve model, λ exhibited positive correlations with P_u, d/L, y/D, and D/L; however, these correlations were relatively modest, with coefficients (ρ) not exceeding 0.20. Conversely, λ demonstrated slight negative correlations with D and L.

For the sand p-y curve model, λ displayed more pronounced positive correlations with y/D and D/L, with ρ values of 0.191 and 0.201, respectively. Negative correlations were observed with φ, d/L, y/L, D, and L. Notably, the correlation coefficients for D and L fell below −0.2, indicating that pile diameter and pile length significantly influence the sand p-y curve behavior. From an engineering standpoint, the strong negative correlation with pile diameter (D) implies that the accuracy of the sand p-y curve model is particularly sensitive to this parameter. This underscores the importance of precise pile diameter specification and manufacturing control in design and construction, as variations can lead to considerable deviations in the predicted soil–pile interaction. Similarly, the significant influence of pile length (L) highlights its critical role in the overall response, demanding careful optimization during the design process.

In the pile horizontal displacement model, F, l/L, and D exhibited positive correlations with λ. Negative correlations were identified with φ, E, t/D, D/L, and L. Particularly, parameters E and L had ρ values less than −0.2, suggesting that both the design parameters of the pile and the stiffness of the soil are significant factors affecting its horizontal displacement. The engineering significance of the strong negative correlation with soil modulus (E) is substantial; it emphasizes that accurate determination of soil stiffness through thorough geotechnical investigation is paramount for reliable predictions of horizontal pile displacement. Even minor uncertainties or variations in E can disproportionately affect the model’s output, potentially impacting the safety and serviceability of the pile foundation. Likewise, the pile length (L), a fundamental design choice, also demonstrates a strong influence, reinforcing the need for its careful selection based on site conditions and load demands.

5.3. Probability Distribution

In reliability-based geotechnical design, the probability distribution of the λ, in addition to its μ_λ and COV_λ, constitutes a critical consideration. Kolmogorov–Smirnov (K–S) normality tests were performed on both the λ and its natural logarithm ln(λ), revealing that the model factors generated by the neural network deviated significantly from normal or log-normal distributions. Further goodness-of-fit tests, including the corrected K-S test and Anderson–Darling (A-D) test, demonstrated that the model factor also failed to conform to Weibull, exponential, or gamma distributions.

To characterize the distribution pattern of the λ, a second-order Gaussian function was employed; this approach proves particularly effective in capturing the highly nonlinear distribution characteristics of model factors [83]. Figure 11 illustrates the fitting results, while

Table 4. Gaussian fitting expression and parameters.

Model	Equation	Parameter	Value	R2
Clay p-y curve	${\displaystyle \sum _{k=1}^{k=2}{a}_k}\exp \left[-{\left(\frac{z-b_k}{c_k}\right)}^2\right]$	a1	790.2102	0.97
		b1	11.5655
		c1	3.90184
		a2	0.94124
		b2	0.08995
		c2	1.71666
Sand p-y curve		a1	0.75607	0.99
		b1	−0.617
		c1	2.0571
		a2	169.8418
		b2	13.26167
		c2	5.31597
Pile horizontal displacement		a1	2.12399	0.99
		b1	3.15144
		c1	2.16728
		a2	0.82494
		b2	−0.5293
		c2	1.68297

details the mathematical expression of the second-order Gaussian function and its optimized parameters. The coefficient of determination values approaching unity across all three investigated scenarios confirm the rationality and efficacy of utilizing Gaussian functions to model the distribution trends of the model factor λ.

To provide a more intuitive comparison of the accuracy among different predictive models, the μ_λ and the COV_λ for each model are summarized in

Table 5. The degree of dispersion and accuracy of the prediction methods of API, ANN, and FEM.

Model	μλ	COVλ
API standard clay p-y curve	1.51	0.73
API standard sand p-y curve	0.70	0.79
ANN clay p-y curve	1.02	0.45
ANN sand p-y curve	1.06	0.29
FEM horizontal displacement	1.41	0.40
ANN horizontal displacement	0.99	0.26

. A comparative analysis reveals that the artificial neural network (ANN) model exhibits significant advantages in predictive accuracy over the API specification and the finite element model. Specifically, the average model factor of the ANN model is significantly closer to 1, and the relatively low coefficient of variation further validates the remarkable stability and reliability of the ANN model during the prediction process. In stark contrast, the FEM and the API approach, in practical applications, highly rely on precise parameter settings and simplified assumptions. When faced with complex and variable actual working conditions, it is difficult to comprehensively capture the subtle differences in the real environment, resulting in a significant reduction in the stability and reliability of their prediction results.

5.4. Case Application

To further investigate the neural network model’s performance, a case study was conducted using a monopile foundation from [84] work. It is critical to emphasize that the monopile analyzed here was excluded from the dataset used for prior model development and validation. The steel pipe pile has a length of 71.5 m, diameter of 2.0 m, and wall thickness of 30 mm. The surrounding soil strata consist of fine sand, silty sand, and clayey silt, with geotechnical properties detailed in

Table 6. Design parameters for the example monopile.

Soil	Soil Thickness (m)	Elasticity Modulus (MPa)	Poisson Ratio	Cohesion (kPa)	$\mathbf{Friction}\mathbf{Angle}(°)$	Unit Weight (kN/m3)
Sedimentary	11.1	18.0	0.3	15.5	16.5	18.4
Alluvial	25.7	20.1	0.3	15.0	17.0	18.8
Fine sand	13.9	36.0	0.2	8.0	33.4	19.5
Silt	20.7	30.3	0.2	24.6	30.5	19.0
Clay	14.5	18.5	0.3	26.2	18.3	20.0

.

Lateral displacement predictions along the pile depth were generated using both FEM and ANN models. As demonstrated in Figure 12, under five distinct loading scenarios, the FEM model consistently underestimated displacements, particularly in the supralittoral zone (above the mudline). This systematic bias correlates with the model accuracy evaluation outcomes, where the μ_λ > 1 (refer to Figure 10). Such underestimation could lead to non-conservative design outcomes in engineering practice, potentially increasing operational risks for monopile foundations. Conversely, the ANN model predictions exhibit significantly better alignment with field monitoring data.

Progressive refinement of the ANN model is achievable through continuous accumulation of field measurements, involving both architectural optimization (e.g., hidden layer configuration) and parameter calibration (e.g., activation function tuning). This iterative process enhances predictive capability while reducing output variance, thereby enabling more robust soil–pile interaction analysis for offshore foundation design. The ANN’s superior predictive accuracy compared to conventional FEM methodologies underscores this study’s contribution to advancing precision in geotechnical predictive modeling.

6. Conclusions

This study focuses on offshore wind turbine monopile foundations. By collecting data from literature sources, we established a marine pile–soil interaction database comprising 1852 datasets. Based on this database, predictive neural network models were developed for pile displacement and soil resistance around the pile, using pile design parameters, displacements, and soil strength parameters as input variables. The accuracy of these models was quantitatively evaluated, and their validity was verified. The main conclusions of this study are as follows:

(i) Artificial neural networks are established as p-y curve models for marine pile foundations in both clay and sand, as well as a predictive model for pile horizontal displacement. The developed pile–soil interaction ANN models exhibit nearly unbiased average accuracy, with an average error of less than 6%.

(ii) The cumulative distribution function of the model factor in the established neural network models can be approximated using a second-order Gaussian function. This simple expression can be directly applied to the reliability-based design of marine pile foundations.

(iii) Through analysis of practical cases and by comparing the finite element method with the neural network models developed in this study for predicting pile horizontal displacement, the superiority of the proposed ANN models in terms of computational accuracy and efficiency is highlighted.

Building upon the established model, subsequent research should prioritize the development of real-time data assimilation mechanisms for digital twin systems to achieve synchronized integration of monitoring data and numerical simulations, coupled with systematic expansion of engineering databases encompassing diverse marine geological conditions. The implementation of hybrid modeling frameworks that synergize 3D convolutional neural networks with physical principles is expected to significantly enhance predictive capabilities in pile–soil interaction analysis.