Automated regression workflow for interpretable deflection prediction in bio-inspired laminated composite plates

Abstract

This study introduces a powerful automated regression workflow (ARW) for accurately predicting deflection in bio-inspired laminated composite plates using diverse machine learning (ML) algorithms. The ARW significantly automates complex processes like hyperparameter optimization, model training, and performance evaluation, accelerating analytical insights. Six different ML regression models were systematically deployed, achieving an impressive average prediction accuracy, five models exceeding 99%, on a comprehensive finite element-generated data set. Notably, the eXtreme gradient boosting regression (XGBR) model exhibited superior performance (R² = 0.999, MAE = 0.010, RMSE = 0.013) on unseen data. Interpretability analyses using SHapley Additive exPlanations and local interpretable model-agnostic explanations on the optimal XGBR model consistently identified boundary conditions and the ratio of elastic moduli (E₁/E₂) as the most influential factors, followed by the aspect ratio (a/h) and loading type. This work establishes an efficient, accurate, and interpretable framework that accelerates the design and fundamental understanding of these complex composite structures, which can be further applied to numerous applications.

Graphical abstract

Keywords

ARW / interpretable machine learning / bio-inspired laminated composites / finite element analysis

1 Introduction

Since the advent of life, biological systems have evolved complex structural architectures through natural selection and evolutionary processes. These systems typically exhibit hierarchical micro- and nanostructures that integrate stiff, inorganic building blocks with soft, organic matrices or interfaces [1]. Such configurations impart remarkable mechanical properties, including high strength and toughness. These biological design principles have inspired the development of bio-inspired laminated composite structures aimed at mimicking their multifunctional performance [2–4]. However, faithfully replicating the intricate hierarchical organization across nano- and microscales remains a formidable challenge [5].

Layered structural arrangements in biological organisms have long been recognized for their functional significance. Organisms such as nacre-producing mollusks [6] and the dactyl club of mantis shrimp [7], which exhibit a sophisticated hierarchical design, have been a prominent source of inspiration for developing materials with superior mechanical performance. Among these, the dactyl club contains a unique internal spiral pattern, known as a helicoidal architecture [8], commonly referred to as the Bouligand structure [9], is particularly noteworthy. This configuration is defined by a stacking sequence in which each lamina is gradually rotated relative to the previous one, forming an overall helical spring-like structure [10]. Such helicoidal architectures confer enhanced mechanical performance, notably increased stiffness and impact resistance, thereby offering protection against severe mechanical and environmental loads [11]. For instance, the forewings (elytra) of certain beetle species exhibit puncture resistance up to 23 N, primarily attributed to the presence of helicoidally arranged chitin fibers [12,13].

The discovery of the Bouligand structure in biological systems, along with its exceptional mechanical performance, has motivated extensive research into the potential advantages of helicoidal laminates over conventional laminate configurations [14]. Grunenfelder et al. [15] replicated the helicoidal architecture of the stomatopod dactyl club in the design of carbon fiber/epoxy composites, demonstrating enhanced impact resistance and toughness. Their results indicated that, under impact loading, helicoidal laminates exhibited significantly reduced damage propagation compared to traditional unidirectional and cross-ply laminates. Similarly, Cheng et al. [10] reported notable improvements in the mechanical properties of bio-inspired helicoidal laminates, particularly with small inter-ply rotation angles, surpassing the performance of standard composite architectures. In another study, Chew et al. [16] evaluated the impact response of helicoidal composite laminates with varying pitch angles (6° and 12°), and observed superior resistance to fiber damage relative to cross-ply and quasi-isotropic laminates. More recently, Yang and Xie [17] demonstrated that Bouligand-type helicoidal architectures significantly enhance the thermal buckling resistance of carbon fiber-reinforced polymers. Sharma et al. [18] provided an in-depth review of the mechanics of bio-inspired helicoidal laminated structures. Furthermore, Ha and Lu [19] presented a comprehensive review on bio-inspired structures for energy absorption, while Budholiya et al. [20] focused on their application in the aerospace industry.

To investigate the mechanical behavior of laminated composite plates, researchers have employed a range of analytical, semi-analytical, and numerical techniques. These approaches have significantly contributed to understanding the structural response and optimizing the design of such materials. For instance, Zhang et al. [21] analyzed the bending resistance and toughness of bio-inspired composites using classical laminate plate theory integrated within a finite element method framework. Mohamed et al. [14] applied the differential quadrature method within the first-order shear deformation theory (FSDT) framework to study the bending, free vibration, and buckling behavior of helicoidal laminated composite plates. In a subsequent study, Mohamed et al. [22] explored the nonlinear bending and snap-through responses of helicoidal composite beams utilizing Bernstein polynomial-based approximations.

The aforementioned studies predominantly employed FSDT, which assumes a constant transverse displacement through the thickness and, consequently, yields a uniform distribution of transverse shear stresses. This simplification often leads to inaccuracies, particularly in thick and layered composite structures. To address these limitations, Garg et al. [23] adopted the higher-order zigzag theory (HOZT) to conduct buckling and free vibration analyses of bio-inspired helicoidal laminated sandwich plates. Subsequently, the authors [24] extended the application of HOZT to the bending analysis of helicoidal laminated composite plates, capturing more accurate through-thickness stress variations. Sharma et al. [18] further utilized a Navier solution-based shear deformation theory to investigate the bending response of bio-inspired helicoidal laminated composite plates. In a complementary effort, Sharma, Shukla et al. [25] employed HOZT to analyze the buckling behavior of both cross-helicoidal and double-helicoidal laminated composite plates. More recently, Kiran et al. [26] implemented an inverse hyperbolic shear deformation theory within a non-uniform rational b-spline based isogeometric analysis (IGA) framework to examine the vibrational and buckling characteristics of bio-inspired laminated composite structures. This approach facilitated smooth geometric representation and higher-order continuity, enabling accurate modeling of complex geometric configurations as highlighted in the works of Timon and coworkers [27,28]. Moreover, the IGA-based approaches also facilitate the adaptive mesh refinement, which can reduce the computational cost by refining the mesh near the defects and cracks in the structures [29,30].

Numerical methods have proven effective in analyzing the mechanical behavior of laminated composite structures. However, such methods often involve substantial computational costs and suffer from mesh sensitivity, particularly in the context of complex architectures such as bio-inspired composites. These limitations become more pronounced in parametric studies or optimization tasks that require repeated simulations. To overcome these challenges, machine learning (ML) techniques have emerged as powerful and computationally efficient alternatives, offering rapid, accurate, and interpretable predictions through data-driven surrogate modeling.

In surrogate modeling, ML algorithms learn complex nonlinear mappings between input parameters and system responses, effectively bypassing the need for repeated high-fidelity simulations. This paradigm is increasingly being adopted in the engineering community for investigating the mechanical behavior of engineering structures. Recent developments in the integration of ML algorithms for structural analysis and materials design can be referred to Refs. [31–33]. These approaches offer a promising alternative to traditional discretization-based methods, providing robust, efficient, and scalable solutions that significantly accelerate the design and analysis processes. In this context, a wide array of ML algorithms, such as artificial neural networks-Gaussian process regression (GPR), linear regression, support vector regression (SVR), gradient boosting regression (GBR), random forest regression (RFR), eXtreme gradient boosting regression (XGBR), and neural network regression (NNR), have been successfully applied for surrogate-based predictions [34]. Recent developments include stochastic multiscale modeling for composite materials, often leveraging hybrid ML algorithms and surrogate models [35,36]. An innovative approach integrating interpretable stochastic ML with multiscale analysis has also been proposed to accurately predict the thermal conductivity of graphene-based polymer nanocomposites [37].

Balcıoğlu and Seçkin [38] demonstrated that regression-based ML models can achieve high accuracy in predicting fracture behavior in composites, outperforming traditional finite element simulations in computational efficiency. Mishra et al. [39] employed Minimax Probability Machine Regression and Multivariate Adaptive Regression Splines to accurately predict buckling loads in skew laminated composite plates. Sikdar and Pal [40] utilized ML models for damage identification, specifically targeting debonding in sandwich composite structures. Kahev et al. [41] applied ML techniques to predict the ultimate buckling loads of variable stiffness composite cylinders with remarkable accuracy. Other applications also involve web-based expert systems for computationally expensive materials design problems [42] and predicting the bending strength of pipeline materials using ML [43].

Further advancing the field, Garget al. [44] utilized GPR to predict the elemental stiffness matrix in functionally graded nanoplates, reducing the computational burden associated with conventional simulations. Mukhopadhyay et al. [45] used GPR to establish a computational bridge between FSDT and HOZT, facilitating efficient multi-theory analysis. Wu et al. [46] applied SVR for the prediction and optimization of the vibration characteristics of bistable composite shells. Additionally, Huang et al. [47] developed a data-driven framework to accurately predict the free vibration behavior of axisymmetric shell structures. More recently, other than data-driven approaches, scientific ML-based approaches for solving solid mechanics problems have gained significant attention from researchers [48,49]. These also encompass explainable Artificial Intelligence (AI) frameworks for material design and engineering applications [50], data-driven quantitative analysis for complex engineering systems [51], stochastic multiscale modeling with global sensitivity analysis for composite materials [52], and multi-scale modeling of thermal conductivity in polyurethane composites using physics-informed neural networks (PINNs) [53]. Recently, a novel PINNs framework was developed by Minh et al. [54] for the static behavior analysis of orthotropic plates. However, the present model is solely dependent on data, which is motivated by the ability to effectively learn complex, high-dimensional relationships directly from simulation data without the need for explicit physical formulations. Data-driven models are particularly advantageous when the governing physics are difficult to define analytically or computationally expensive to solve, and they offer rapid predictions once trained, beneficial for tasks such as surrogate modeling and design optimization [55,56].

The paper is structured as follows. Section 2 provides an overview of mathematical formulation, including the displacement field, strain–displacement relationship, constitutive relationship, and variational formulation. The problem validation and the parameters considered for database generation for training ML models are discussed in Section 3. The details of the automated regression workflow (ARW) developed and different ML regression models employed are provided in Section 4. A discussion on the performance of the ML regression models and their interpretability is present in Section 5. The conclusions in Section 6 summarize the key findings and potential implications for future research. An overview of the work is presented in Fig. 1 below.

2 Mathematical formulation

The subsequent sections establish the mathematical equations for the displacement field and the strain–displacement relationship used for the analysis of bioinspired laminated composite plates.

2.1 Displacement field

For a bio-inspired laminated composite plate, having length a, width b, and thickness h, as shown in Fig. 2 the displacement field is given as follows, using the third-order shear deformation theory [57]

(1a)\begin{array}{c}u\left(x,y,\mathrm{z}\right)=u_0+\mathrm{z}{\beta }_x+c{\mathrm{z}}^3\left({\beta }_x+\frac{\mathrm{\partial }w}{\mathrm{\partial }x}\right),\end{array}

(1b)\begin{array}{c}v\left(x,y,\mathrm{z}\right)=v_0+\mathrm{z}{\beta }_y+c{\mathrm{z}}^3\left({\beta }_y+\frac{\mathrm{\partial }w}{\mathrm{\partial }y}\right),\end{array}

(1c)\begin{array}{c}w\left(x,y\right)=w_0.\end{array}

In this formulation, u, v, and w represent the displacement components in the x, y, and z directions at any point within the plate. The terms u₀, v₀, and w₀ denote the corresponding displacements of the mid-surface of the plate. The angular rotations β_x and β_y correspond to the rotations of the transverse normal about the y-axis and the x-axis, respectively. The constant c, defined as c = 4/3h², serves as a shear correction factor.

2.2 Strain–displacement relationship

The present study considers linear strain theory under the assumptions of small deformations and small rotations. Accordingly, the in-plane strain vector {\mathit{\epsilon }}_{\mathrm{p}} can be expressed as

(2a)\begin{array}{c}{\mathit{\epsilon }}_{\mathrm{p}}={[\begin{array}{ccc}{\epsilon }_{xx}& {\epsilon }_{yy}& {\gamma }_{xy}\end{array}]}^{\mathrm{T}}={\mathit{\epsilon }}_0+\mathrm{z}{\mathit{\varkappa }}_1+{\mathrm{z}}^3{\mathit{\varkappa }}_2,\end{array}

where

(2b)\begin{array}{rl}& {\mathit{\epsilon }}_0=\left[\begin{array}{c}{\displaystyle \frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }x}}\\ {\displaystyle \frac{\mathrm{\partial }{v}_0}{\mathrm{\partial }y}}\\ {\displaystyle \frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }y}}+{\displaystyle \frac{\mathrm{\partial }{v}_0}{\mathrm{\partial }x}}\end{array}\right],{\mathit{\varkappa }}_1=\left[\begin{array}{c}{\displaystyle \frac{\mathrm{\partial }{\beta }_x}{\mathrm{\partial }x}}\\ {\displaystyle \frac{\mathrm{\partial }{\beta }_y}{\mathrm{\partial }y}}\\ {\displaystyle \frac{\mathrm{\partial }{\beta }_x}{\mathrm{\partial }y}}+{\displaystyle \frac{\mathrm{\partial }{\beta }_y}{\mathrm{\partial }x}}\end{array}\right],\\ & {\mathit{\varkappa }}_2=\left[\begin{array}{c}{\displaystyle \frac{\mathrm{\partial }{\beta }_x}{\mathrm{\partial }x}}+{\displaystyle \frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }{x}^2}}\\ {\displaystyle \frac{\mathrm{\partial }{\beta }_y}{\mathrm{\partial }y}}+{\displaystyle \frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }{y}^2}}\\ {\displaystyle \frac{\mathrm{\partial }{\beta }_x}{\mathrm{\partial }y}}+{\displaystyle \frac{\mathrm{\partial }{\beta }_y}{\mathrm{\partial }x}}+2{\displaystyle \frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }x\mathrm{\partial }y}}\end{array}\right].\end{array}

The transverse shear strain vector \mathit{\gamma } is given as

(3a)\begin{array}{c}\mathit{\gamma }={[\begin{array}{cc}{\gamma }_{x\mathrm{z}}& {\gamma }_{y\mathrm{z}}\end{array}]}^{\mathrm{T}}={\mathit{\epsilon }}_{\mathrm{s}}+{\mathrm{z}}^2{\mathit{\varkappa }}_{\mathrm{s}},\end{array}

where

(3b)\begin{array}{c}{\mathit{\epsilon }}_{\mathrm{s}}=\left[\begin{array}{c}{\beta }_x+{\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }x}}\\ {\beta }_y+{\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }y}}\end{array}\right],{\mathit{\varkappa }}_{\mathrm{s}}=3c\left[\begin{array}{c}{\beta }_x+{\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }x}}\\ {\beta }_y+{\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }y}}\end{array}\right].\end{array}

2.3 Constitutive relationship

The constitutive relationship of a kth orthotropic lamina is governed by Hooke’s law, assuming the plane stress condition, and is given as

(4){\left\{\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\tau }_{xy}\\ {\tau }_{x\mathrm{z}}\\ {\tau }_{y\mathrm{z}}\end{array}\right\}}^{\left(k\right)\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}={\left[\begin{array}{ccccc}{Q}_{11}& Q_{12}& Q_{16}& 0& 0\\ Q_{21}& Q_{22}& Q_{26}& 0& 0\\ Q_{61}& Q_{62}& Q_{66}& 0& 0\\ 0& 0& 0& Q_{55}& Q_{54}\\ 0& 0& 0& Q_{45}& Q_{44}\end{array}\right]}^{\left(k\right)}{\left\{\begin{array}{c}{\epsilon }_{xx}\\ {\epsilon }_{yy}\\ {\gamma }_{xy}\\ {\gamma }_{x\mathrm{z}}\\ {\gamma }_{y\mathrm{z}}\end{array}\right\}}^{\left(k\right)\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}},

where Q_ij are computed as

\begin{array}{rl}& Q_{11}=\frac{E_{11}}{1-{\vartheta }_{12}{\vartheta }_{21}},Q_{12}=\frac{{\vartheta }_{12}{E}_{22}}{1-{\vartheta }_{12}{\vartheta }_{21}},Q_{22}=\frac{E_{22}}{1-{\vartheta }_{12}{\vartheta }_{21}},\\ & Q_{66}=G_{12},Q_{55}=G_{13},Q_{44}=G_{23}.\end{array}

A laminate generally comprises multiple orthotropic layers, each oriented at a specific angle with respect to the global laminate coordinate system. The stress–strain relationship for the kth layer is expressed as follows in global coordinate system

(5){\left\{\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\tau }_{xy}\\ {\tau }_{x\mathrm{z}}\\ {\tau }_{y\mathrm{z}}\end{array}\right\}}^{\left(k\right)\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{l}}={\left[\begin{array}{ccccc}{\overline{Q}}_{11}& {\overline{Q}}_{12}& {\overline{Q}}_{16}& 0& 0\\ {\overline{Q}}_{21}& {\overline{Q}}_{22}& {\overline{Q}}_{26}& 0& 0\\ {\overline{Q}}_{61}& {\overline{Q}}_{62}& {\overline{Q}}_{66}& 0& 0\\ 0& 0& 0& {\overline{Q}}_{55}& {\overline{Q}}_{54}\\ 0& 0& 0& {\overline{Q}}_{45}& {\overline{Q}}_{44}\end{array}\right]}^{\left(k\right)}{\left\{\begin{array}{c}{\epsilon }_{xx}\\ {\epsilon }_{yy}\\ {\gamma }_{xy}\\ {\gamma }_{x\mathrm{z}}\\ {\gamma }_{y\mathrm{z}}\end{array}\right\}}^{\left(k\right)\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{l}},

where {\overline{Q}}_{ij} represents the transformed material constant and can be referred to Ref. [32].

2.4 Variational formulation

The present study employs Hamilton’s principle for deriving the generalized governing equations for the laminated structures. Accordingly, the governing equations for the static analysis can be found using Hamilton’s principle for the given system, which is expressed as follows

(6)\delta {\int }_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}L\mathrm{d}t={\int }_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}(\delta U+\delta W_{\mathrm{e}\mathrm{x}\mathrm{t}})\mathrm{d}t.

where U and W_{\mathrm{e}\mathrm{x}\mathrm{t}} refer to strain energy and external work done, while \delta represents the variational operator. The virtual strain energy stored in the bioinspired composite plate is expressed as

(7)\delta U={\int }_V^{}\delta {\left\{\mathit{\epsilon }\right\}}^{\mathrm{T}}\left\{\mathit{\sigma }\right\}\mathrm{d}V.

The first variation of the work done by the transverse mechanical load P is expressed as

(8){\delta W}_{\mathrm{e}\mathrm{x}\mathrm{t}}={\int }_A^{}\delta w^{\mathrm{T}}P\mathrm{d}A,

where A denotes the elemental area.

Furthermore, the plate geometry is discretized using isoparametric Lagrangian finite elements, connected through nodes that have seven degrees of freedom. For the sake of brevity, the authors choose to omit the details of finite element discretization; however, the necessary details can be found in Ref. [58]. Eventually, substituting Eqs. (7) and (8) in Eq. (6) and using the finite element discretization, the governing equation for the bending analysis of bioinspired composite plate under transverse load is obtained as

(9)\left[\mathit{K}\right]\left\{\mathit{q}\right\}={\{\mathit{F}}_{\mathrm{m}}\},

where \left[\mathit{K}\right] represents the global stiffness matrix, \mathit{q} refers to the global nodal displacement vector and {\{\mathit{F}}_{\mathrm{m}}\} represents the global mechanical load vector.

3 Problem validation and database generation

This section presents comparative studies with existing literature to validate the present finite element approach. As a starting point, the framework is validated to investigate the bending response of standard composite laminates. Thereafter, the framework is employed to generate a comprehensive database investigating the bending response of bioinspired composites with different layups under different boundary and loading conditions. In the present study, until stated otherwise, the following material properties are used for the analysis purpose, which are in line with those presented in Ref. [31]

(10)\frac{E_1}{E_2}=\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n},G_{12}=G_{13}=0.5E_2,G_{23}=0.2E_2,{\nu }_{12}=0.25.

Also, the transverse loading P(x, y) is described as follows

(11)\begin{array}{rl}& P\left(x,y\right)\\ =& \{\begin{array}{lr}{P}_0\sin \left({\displaystyle \frac{\pi x}{a}}\right)\sin \left({\displaystyle \frac{\pi y}{b}}\right),& \mathrm{S}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{s}\mathrm{o}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}\left(\mathrm{S}\mathrm{S}\mathrm{L}\right),\\ P_0,& \mathrm{U}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{l}\mathrm{y}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}\left(\mathrm{U}\mathrm{D}\mathrm{L}\right),\end{array}\end{array}

where P_0 is the intensity of the transverse load. Subsequently, the computed central deflection w\left({\displaystyle \frac{a}{2}},{\displaystyle \frac{b}{2}}\right) is expressed in terms of nondimensional form as

(12)\overline{w}=w\frac{100E_2{h}^3}{a^4{P}_0}.

The present approach has been validated for the bending analysis of simply supported square laminated composites with a configuration of [0°/90°/90°/0°] and [0°/90°/0°/90°/0°/90°/]. All the laminae are considered to be of equal thickness, which is a quarter of the total thickness h. The orthotropic material properties for the lamina are given as E₁ = 25E₂, G₁₂ = G₁₃ = 0.5E₂, G₂₃ = 0.2E₂, and ν₁₂ = 0.25. The plates are subjected to a sinusoidal distributed load and central deflection for three thickness-to-width ratios h/b, including 0.25, 0.1, and 0.01, are compared with the values given by Reddy [57] in Table 1.

To ensure the efficacy of advanced ML algorithms, it is imperative to possess a data set of an adequate size and quality. Consequently, a robust and effective design scheme of numerical experiments based on the finite element simulations has been used to generate a representative data set. The present study investigates the bending analysis of bioinspired laminates with configurations including helicoidal recursive (HR1, HR2, HR3), helicoidal exponential (HE1, HE2, HE3), helicoidal semicircular (HS1, HS2, HS3), linear helicoidal (LH1, LH2, LH3), Fibonacci helicoidal (FH), and quasi-isotropic (QI). The layouts and stacking sequences of these configurations are presented in Refs. [14,59] and Table 2, respectively. The other input features include elasticity ratio E₁/E₂ (10 and 40), types of loading including Uniformly Distributed Load (UDL) and Sinusoidal Load (SSL), constraints or boundary conditions including Simply Supported (SSSS) and Clamped (CCCC), length-to-thickness ratio a/h (10 and 100), and number of layers (12, 16, and 20). With the variation of all the input features, an exhaustive database with 672 datapoints was generated and is employed to train different ML algorithms. The selected input parameter values are aimed at thoroughly exploring their impact on the deflection characteristics of laminated composite plates across an extended parameter space, as they directly influence the structural stiffness, anisotropy, and boundary response characteristics. Among the above-listed features, the configurations, types of loading, and boundary conditions were one-hot encoded as they are categorical variables. Moreover, the QI configuration, SSSS boundary condition, and UDL loading type were eliminated from the features after one-hot encoding as they are dependent on their complementary variables. Additionally, no stochasticity in material properties or geometry has been considered in the present study.

4 Machine learning framework methodology

4.1 Automated regression workflow

In this study, a novel ARW, using the Python programming language, was developed to streamline and simplify the building of ML regression models optimized via Bayesian optimization (BO), previously showcased in the reported ML regression workflows [61]. Traditionally, the development and optimization of hyperparameters for multiple regression models are both time-consuming and complex, often requiring extensive manual coding efforts. To address this issue, a highly efficient and user-friendly Python-based automated framework was designed, significantly reducing the complexity involved in building optimized regression models.

The developed ARW efficiently automates approximately 300 lines of detailed Python code required for the complete development, hyperparameter optimization, and evaluation of ML regression models. Remarkably, this extensive automation is condensed into merely 10–20 lines of user-defined code, where the researcher specifies the desired models along with their hyperparameters and respective optimization ranges. By utilizing BO via the Scikit-optimize [62] library, the ARW systematically identifies optimal hyperparameters, enhancing model accuracy and reliability, focusing on maximizing coefficient of determination (R²) of the model.

The hyperparameter optimization strategy integrated into ARW includes a robust cross-validation process with 5-fold cross-validation to ensure reliable and generalized model performance. Additionally, the workflow incorporates a systematic train–test data split (80–20) for model training and performance evaluation, ensuring unbiased and robust assessments of predictive accuracy.

A core feature of this automated workflow is its intuitive design, promoting ease of use even for researchers with minimal programming experience. To implement ARW, users simply need to store the provided Python script (mlregworkflow.py) within their working directory and execute the following concise import command: from mlregworkflow import run_workflow.

Researchers are required to import their chosen regression models from available ML libraries (e.g., Scikit-learn [63]) and explicitly specify the hyperparameters along with their respective value ranges for optimization. ARW subsequently automates the optimization and evaluation process.

ARW also automatically generates diagnostic visualizations, such as predicted versus true value plots for both training and testing data sets, for each optimized regression model. These visualizations are stored in a default folder named plots within the working directory. Furthermore, ARW outputs a detailed CSV file (results.csv) containing the model’s name, optimized hyperparameters, and performance metrics for both the training and testing data sets. The performance metrics evaluated are R², mean absolute error (MAE), and root mean squared error (RMSE). These plots and the CSV file serve as essential resources for assessing model performance, providing immediate insights into predictive accuracy, and aiding in model selection.

To demonstrate its practical utility, the ARW was tested extensively across various regression models, consistently showcasing improved performance metrics. Additionally, the automated generation of detailed prediction plots significantly enhances interpretability and validation of model predictions.

The developed ARW thus represents a robust, highly accessible, and efficient tool that significantly accelerates and simplifies the regression modeling process. It not only reduces manual coding workload but also promotes broader applicability and reproducibility of optimized ML models across diverse research disciplines.

4.2 Predictive regression models and interpretable machine learning

Six different regression models are developed to accurately map input features to the deflection values of bio-inspired laminated composite plates. These models included linear regression (Linear), SVR, GBR, RFR, XGBR, and NNR. The primary objective of employing these diverse ML-based models was to rigorously analyze and compare their predictive capabilities regarding the deflection behavior of various configurations of bio-inspired laminated composite plates under different boundary conditions and loading scenarios.

The hyperparameter optimization for these models followed the same systematic approach previously outlined in the ARW subsection. Linear regression did not require hyperparameter tuning and thus was excluded from this optimization process.

Table 3 details the optimized hyperparameters and their respective value ranges utilized for each ML regression model. Notably, the NNR had fixed epochs and batch-size parameters, set at 100 and 32, respectively.

To evaluate the prediction accuracy and quantify the errors associated with each regression model, three key metrics are computed: MAE, RMSE, and R², defined by the following equations:

(13)MAE=\frac{1}{n}\sum _{i=1}^n\left|y_i-{\hat{y}}_i\right|,

(14)RMSE=\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(y_i-{\hat{y}}_i\right)}^2},

(15)R^2=1-\frac{{\displaystyle \sum _{i=1}^n{\left(y_i-{\hat{y}}_i\right)}^2}}{{\displaystyle \sum _{i=1}^n{\left(y_i-{\overrightarrow{y}}_i\right)}^2}},

where n is the sample size, y_i is the actual output value, and {\hat{y}}_i is the predicted output value.

From the evaluation of these metrics, the best-performing model, exhibiting the highest R² and lowest MAE and RMSE, was selected for further interpretation. To gain deeper insights into how individual features influenced the deflection behavior of the bio-inspired laminate composites, advanced interpretability analyses using SHapley Additive exPlanations (SHAP) via the SHAP library and local interpretable model-agnostic explanations (LIME) via the LIME library are performed on the top-performing model.

5 Results and discussion

This section presents the Bayesian-optimized hyperparameters alongside performance metrics (R², MAE, RMSE) for each regression model on the training (80%) and testing (20%) data sets. A systematic comparison of the performances then identified the highest-accuracy model. Finally, SHAP and LIME analyses produced both global explanations, quantifying overall feature-importance patterns, and local explanations, that dissect each feature’s contribution to an individual prediction, to illuminate the drivers of model behavior.

5.1 Predictive regression models

Table 4 summarizes the final hyperparameter settings selected by BO for each regression algorithm. As expected, the Linear model requires no tuning. The SVR model favors an RBF kernel with epsilon = 0.01 and C = 8.747. The regularization parameter C must be strictly positive and controls the trade-off between model flatness and penalization of deviations, penalty strength is inversely proportional to C and uses a squared L₂ norm. The epsilon defines an ‘epsilon-tube’ around the regression function: residuals within ± epsilon incur no loss, while only errors exceeding this tolerance contribute to the training objective. GBR converged on a shallow ensemble of eight trees (n_estimators = 8) with a high learning rate (0.90), squared-error loss, moderate regularization (alpha = 0.124), and tree depth of five (max_depth = 5), reflecting a bias, variance trade-off tuned for this data set. The RFR ensemble uses only three trees of depth six, indicating that even a small forest can capture most variance. XGBR employs ten boosting rounds (n_estimators = 10), a deep tree structure (max_depth = 10), and a learning rate of 0.723 to rapidly fit residuals. Finally, the NNR is configured with a single hidden layer of 128 units, tanh activation, and a learning rate of 7.9 × 10⁻³, combining sufficient representational capacity with controlled complexity.

The Linear model exhibits moderate fitness, as shown in Fig. 3 below, with systematic deviations from the ideal line indicating underfitting in capturing nonlinear trends.

The SVR model achieves near-perfect training performance but shows a slight drop on the test data set, as shown in Fig. 4 below, reflecting strong capacity yet modest overfitting to the training data.

The GBR model attains perfect training alignment and excellent generalization on test data set as shown in Fig. 5 below, demonstrating the power of staged additive trees to capture complex patterns with minimal overfitting.

The RFR model delivers robust ensemble performance, as shown in Fig. 6, balancing bias and variance to yield consistently accurate predictions on unseen data as evident by the tight clustering of the points around the ideal line (red-dotted line).

The XGBR combines perfect fit on training data with near-ideal test data accuracy as shown in Fig. 7, underscoring its efficiency in optimizing both precision and generalization via gradient-boosted trees which can be evidenced by the nearly flawless overlap of predicted and actual values along the ideal line in the both the plots for training and testing data sets.

The NNR model achieves outstanding fidelity on both training and testing data sets as shown in Fig. 8, reflecting its ability to approximate complex nonlinear relationships while maintaining generalization through regularization and controlled architecture.

All the results discussed above on optimized hyperparameter, model building and validation, performance visualization, and metric computation are performed using the ARW, showcasing how it minimizes the effort needed in manual coding. Further, the performance of these six predictive models is compared on the held-out test data set by plotting the R², MAE, and RMSE metrics as shown in Fig. 9 below. The linear model shows the lowest predictive accuracy (R² = 0.881; MAE = 0.104; RMSE = 0.120), whereas all the other models achieve high fidelity (R² ≈ 0.996–0.998; MAE ≈ 0.010–0.015; RMSE ≈ 0.014–0.021). XGBR model outperforms all other models, attaining the highest R² (0.999) alongside the lowest MAE (0.010) and RMSE (0.013), thereby demonstrating superior predictive precision and generalization. Its advanced ensemble boosting technique sequentially builds trees to correct the errors of previous trees, effectively handling complex nonlinearities and interactions within the data set. XGBR’s built-in regularization techniques (e.g., subsample, colsample_bytree, and eta) contribute to its robustness against overfitting, leading to exceptional generalization on unseen data. The specific hyperparameter values found (e.g., max_depth = 10, n_estimators = 10, learning_rate = 0.723) suggest that a balance was achieved between model complexity and learning rate, allowing it to capture intricate patterns without memorizing the training data. Consequently, the XGBR model was selected for subsequent interpretability analyses using SHAP and LIME to elucidate global feature-importance patterns and generate local explanations.

5.2 Interpretable machine learning using SHapley Additive exPlanations and local interpretable model-agnostic explanations analyses on the extreme gradient boosting regression model

Following the identification of XGBR as the top-performing algorithm, this subsection applies SHAP and LIME to unpack its decision logic for predicting the plate deflections. For global interpretation, SHAP summary and beeswarm plots alongside LIME feature importance and beeswarm visualizations are employed to quantify each input’s overall effect on the predicted deflection values. Local interpretation focuses on the first instance of the held-out test set. For this case-specific analysis, SHAP waterfall chart and LIME explanation plot are leveraged to decompose individual deflection predictions into their constituent feature contributions, thereby providing detailed, case-specific insights into model behavior.

5.2.1 SHapley Additive exPlanations analysis

For global SHAP interpretation, two complementary visualizations are employed: the SHAP summary bar chart, which ranks features by their mean absolute SHAP (mean |SHAP|) values to indicate overall importance, and the SHAP beeswarm plot, which displays the distribution of individual SHAP values for each feature, colored by feature value, to reveal both the magnitude and direction of each feature’s impact. Together, these plots offer a comprehensive, model-agnostic perspective on the primary drivers of predicted deflection behavior.

Figure 10 below illustrates both the SHAP summary bar plot and beeswarm plot. The SHAP summary bar chart in Fig. 10(a) ranks the input features by their mean absolute impact on the XGBR model’s deflection predictions. ‘Constraint_CCCC’ (clamped constraint or boundary condition) emerges as the most influential variable, with a mean |SHAP| exceeding 0.60, followed by ‘E₁/E₂’ (~0.58) and ‘a/h’ (~0.30). ‘Loading Type_SSL’ appears next (~0.25), indicating that if the plate is subjected to sinusoidal loading, then it has a non-negligible effect on predicted deflections. All remaining features, including the various configuration encodings and layer count, exert only minor global influence, with mean |SHAP| values near zero. This hierarchy reveals that the model primarily relies on the boundary condition (clamped or simply supported) and elasticity ratio to forecast deflection values.

The corresponding SHAP beeswarm plot in Fig. 10 (b) further elucidates each top feature’s directional effect, where each dot represents one test-set sample and is colored by the corresponding feature value (red = high, blue = low). High values of ‘Constraint_CCCC’ predominantly lie on the negative SHAP side, indicating that an all edges clamped boundary condition reduces the predicted deflection relative to all edges simply supported boundary condition due to enhanced stiffness of the structure. A similar pattern holds for ‘E₁/E₂’, ‘a/h’, and ‘Loading Type_SSL’, where higher feature values consistently correspond to negative SHAP contributions, each acting to suppress deflection in the model’s predictions. Conversely, low feature values for these same parameters push SHAP values positively, driving larger deflection estimates. All remaining configuration encodings and the layer count cluster tightly around zero, confirming their negligible global effect on prediction of deflection values.

For local interpretation using SHAP, the waterfall plot is utilized, which breaks down the model’s output into its baseline value and the individual contributions of each feature, showing how they cumulatively produce the final prediction. Figure 11 shows the waterfall plot (all values shown are on the scaled data) for the first test instance (#0), which decomposes the XGBR prediction of deflection value (f\left(x\right) = −0.804) around the model’s expected value (E[f(x)] = −0.137) into individual feature contributions. SHAP enforces, for any instance x, the exact additive decomposition:

(16)f\left(x\right)=E[f(x)]+\sum _{i=1}^d{\varphi }_i\left(x\right),

where f\left(x\right) is the model’s prediction for the specific feature vector x; E[f(x)] is the baseline or expected model output (i.e., the average f over the data distribution); {\varphi }_i\left(x\right) is the SHAP value for the feature i, representing how much feature i shifts the prediction away from the expected baseline.

The values mentioned in the brackets are scaled values after standardization. The ‘Constraint_CCCC’ feature (value = 1.017) exerts the largest downward push (−0.62), indicating that a fully clamped boundary condition strongly reduces predicted deflection. Conversely, the ‘E₁/E₂’ (−1.009) provides the largest upward contribution (+0.40), partially offsetting the clamp effect by reflecting the influence of a less stiffer material distribution. Subsequently, smaller negative contributions arise from the ‘a/h’ (−0.26) and ‘Loading Type_SSL’ (−0.18), both of which further decrease the deflection value in this configuration. Only the ‘Configuration_LH3’ adds a minor positive effect (+0.02), while all other layer- and configuration-encoding features contribute negligibly (< 0.01) because the configuration of this test instance is LH3. This breakdown highlights how a few dominant mechanical parameters drive the model’s individual deflection prediction for this case. For reference, the actual and predicted scaled deflection values for this instance are −0.817 and −0.804, respectively, corresponding to real deflection values of 0.243 and 0.247 mm.

5.2.2 Interpretable model-agnostic explanations analysis

As described in Subsubsection 5.2.1, the SHAP summary bar chart and beeswarm plot together provide a comprehensive view of how the model predicts deflection behavior. For the LIME analysis, a feature-importance plot is constructed in the same style as the SHAP summary bar chart, and a complementary LIME beeswarm plot is also generated to visualize each feature’s impact across the data set.

Figure 12 below shows the LIME feature importance chart and beeswarm plot. The LIME feature importance bar chart in Fig. 12(a) confirms that ‘Constraint_CCCC’ is the dominant global driver of the XGBR model’s deflection predictions, exhibiting the highest mean absolute LIME ‘E₁/E₂’ weight (~1.25). ‘E₁/E₂’ follows closely (~1.15), indicating that material stiffness distribution is the second most critical factor. The ‘a/h’ (~0.55) and ‘Loading Type_SSL’ (~0.50) are the subsequent significant important factors, while all other configurational encodings and ‘Number of Layer’ register mean weights below 0.10, demonstrating minimal global impact. These results closely mirror the SHAP rankings, confirming that boundary conditions and the elastic‐modulus ratio (E₁/E₂) are the primary drivers of the model’s predictions, while the aspect ratio (a/h) and loading type have a moderate effect, and layer/configuration variables play a minimal role on average.

The corresponding LIME beeswarm plot in Fig. 12(b) provides further insights by showing how individual feature values map to positive or negative contributions. For ‘Constraint_CCCC’, high values (red) consistently produce negative LIME weights, indicating that fully clamped boundaries suppress predicted deflection, while low values (blue) push weights toward positive direction. A similar pattern holds for ‘E₁/E₂’: larger ratios (red) drive negative weights (reducing deflection), whereas smaller ratios (blue) yield positive weights (increasing deflection). The ‘a/h’ and ‘Loading Type_SSL’ features also show this sign-inversion behavior but lesser impactful, with high feature values contributing negatively and low values positively. All other features’ dots cluster tightly around zero, regardless of color, reflecting their negligible global effect on predicted deflection. Thus, the LIME beeswarm clarifies not only which features are important on average, but also how changes in feature magnitude systematically shift the model’s output. Importantly, both LIME and SHAP analyses yield consistent rankings and directional effects, reinforcing the robustness of the global interpretability results.

Figure 13 shows the LIME explanation plot (all values shown are on the scaled data) for the first test instance (#0), offering a detailed, instance-specific breakdown of how feature values drive the XGBR deflection prediction. In the contribution bar chart (left), features pushing the prediction downward (negative LIME weights) are shown in blue, most prominently ‘Constraint_CCCC’ (scaled > −0.98, weight = −1.23), ‘a/h’ (scaled ≤ 1.02, weight = −0.62), and ‘Loading Type_SSL’ (scaled > –1.04, weight = −0.46). The ‘Configuration_HR1’ and ‘Configuration_HR2’ contribute slightly to push the prediction downward with weights −0.13 and −0.1. On the positive side (orange bars), ‘E₁/E₂’ (scaled ≤ −1.01, weight = +1.09) provides the strongest uplift, and ‘Configuration_LH3’ (weight = +0.08) contributes a minor boost which can be attributed to the fact that the configuration of this test instance is LH3. All other configuration and layer features have negligible weights, reflecting their minimal local influence. The gauge situated at top right shows the predicted scaled deflection value of −0.80 within the LIME surrogate’s output range of −1.31 to 2.33, and the accompanying table lists each feature’s actual scaled value, clarifying which numeric inputs generate these weights.

When compared with the SHAP waterfall analysis for the same instance (Fig. 13), both methods identify the same small subset of mechanical parameters as dominant. SHAP shows ‘Constraint_CCCC’ delivering the largest downward push and ‘E₁/E₂’ supplying the principal upward shift, with smaller negative contributions from ‘a/h’ and ‘Loading Type_SSL.’ LIME’s weights differ slightly in magnitude but showcases similar direction and relative ordering of these effects, and both techniques agree that all other features, especially layer/configuration encodings, exert virtually no influence on the individual prediction, aside from the one active configuration feature. Together, these consistent local interpretations confirm that the XGBR model’s single-case decision is driven primarily by boundary condition and elasticity ratio parameters, providing robust, complementary insights into the model’s inner workings.

6 Conclusions

This research successfully developed and validated a novel ARW that significantly streamlines the process of building, optimizing, and interpreting ML models for predicting the deflection of bio-inspired laminated composite plates. ARW’s capability to automate complex modeling tasks, from hyperparameter optimization to detailed performance metrics and visualizations, proved invaluable in enhancing efficiency and reproducibility. The key conclusions drawn from this comprehensive study are as follows.

1) The novel ARW effectively automates the entire regression modeling pipeline, significantly enhancing efficiency and reproducibility in ML model development.

2) It provides a robust, user-friendly platform, minimizing manual coding efforts and accelerating the research cycle.

3) Six diverse ML regression models were rigorously trained and evaluated on a 672-datapoint data set, demonstrating exceptional predictive capabilities for plate deflection.

4) The XGBR model achieved superior performance (R² = 0.999, MAE = 0.010, RMSE = 0.013), establishing it as a high-fidelity surrogate model.

5) SHAP and LIME interpretability analyses on the optimal XGBR model provided critical insights into feature importance.

6) Boundary conditions (specifically clamped supports) and the ratio of elastic moduli (E₁/E₂) are the primary drivers of deflection.

7) The aspect ratio (a/h) and loading type also showed moderate influence on predictions.

8) These insights offer crucial understanding for optimizing bio-inspired composite design parameters.

This study delivers a powerful, accurate, and transparent ML framework facilitated by the ARW, which holds substantial promise for advancing the analysis and design of complex composite structures. The real-world applications span the aerospace, marine, and automotive sectors, where timely and high-fidelity prediction of deformation in lightweight composite panels is crucial, especially during the initial phases of structural design. Moreover, this approach is valuable in the renewable energy domain, including wind turbine blades and solar panel supports, where controlling deflection characteristics is vital for optimizing functionality and longevity. The ability to rapidly predict coupled with insightful interpretation of mechanical behaviors offers distinct advantages for materials science and mechanical engineering.

Future work could explore broader applications of this framework, including its integration with experimental studies, incorporation into multi-physics simulations, and extension to address various material behaviors and structural integrity challenges. Further, the framework can be extended to assess the global structural response under the influence of defects and cracks. Additionally, it could also integrate uncertainty quantification methods, such as confidence intervals on predictions or the use of model ensembles, to provide a more comprehensive assessment of prediction reliability. Furthermore, incorporating advanced sensitivity analysis techniques would offer a clearer understanding of the individual and interactive importance of various input parameters, thereby enhancing the robustness and applicability of the framework. Moreover, future versions of the model could benefit from integrating physics-based constraints via AI for partial differential equations techniques, offering a more principled framework for incorporating domain knowledge. This could involve, for instance, the exploration of PINNs, which can embed physical constitutive equations directly into the ML framework, potentially enhancing generalizability and interpretability by ensuring adherence to fundamental physical laws.

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