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Abstract

This study presents a machine learning (ML) framework for predicting the critical buckling loads of tapered axially functionally graded (AFG) nanobeams resting on elastic foundations, governed by Eringen’s nonlocal elasticity theory. Due to their position-dependent stiffness and complex boundary conditions, closed-form solutions for such nanobeams are computationally expensive and analytically intractable. To address this, four supervised data-driven regression models are developed: linear regression (LR), polynomial regression (PR), support vector machine (SVM), and artificial neural network (ANN). These models are used to predict buckling loads for three boundary conditions: clamped–clamped (C–C), simply supported (S-S), and clamped-simply supported (C–S). A comprehensive dataset is generated using a finely discretized parametric sweep of the beam’s geometric and material properties. Pearson correlation coefficient (PCC) and variance inflation factor (VIF) analyses are conducted to ensure feature relevance and independence. Results show that the ANN model consistently outperform other test models, achieving coefficient of determination (R²) values exceeding 99%, root mean square error (RMSE) values below 1.1 (in normalized units), and mean absolute error (MAE) values below 0.67 (in normalized units) across all boundary conditions. The findings demonstrate that ML and ANN offer a robust and computationally efficient surrogate to traditional methods, with strong generalization capabilities.

Keywords

critical buckling load AFG nanobeams nonlocal elasticity theory machine learning artificial neural networks

Introduction

Nanobeams are fundamental load-bearing members in many micro- and nano-electro-mechanical systems and flexible electronics platforms.¹ Because their slender geometry makes elastic instability more critical than material strength, the critical buckling load is the primary design metric.² To tune stiffness, weight, and multifunctional response, designers increasingly use axially functionally graded (AFG) materials together with deliberate geometric tapering along the span.^3,4 The combined variation of material moduli, density, and cross-section converts the bending rigidity into a spatially varying quantity, which violates the constant-coefficient assumptions behind classical Euler–Bernoulli theory.⁵ Recent elastostatic work on tapered functionally graded beams shows that the combined taper and spatially varying material properties lead to position-dependent stiffness coefficients, so specialized polynomial-coefficient finite-element formulations are required instead of simple closed-form solutions.⁶

At nanoscale dimensions, non-local stress interactions bring in characteristic length parameters that must be carried through the governing equations.⁷ Large-scale molecular-dynamics benchmarks confirm that size-dependent stiffness and critical load can diverge markedly from classical predictions, highlighting the need for enriched continuum models.⁸

Multiscale analyses on carbon-nanotube-reinforced laminates show that environmental conditions and microstructural imperfections reduce the critical buckling load.⁹ Studies on asymmetric composite sandwich panels with tapered regions likewise find that geometric asymmetry and material heterogeneity cause shear-buckling damage and matrix-splitting cracks, with ultimate load dictated by core-thickness-dependent stiffness.¹⁰ Because these mechanisms must be resolved in detail, high-fidelity finite-element or mesh-free models become costly. A recent butt-joint stiffened thermoplastic composite panel analysis demanded very fine meshing and nonlinear solvers merely to converge,¹¹ and mapping the influence of fastener layout and shear-connector stiffness on built-up cold-formed columns required 84 separate finite-element runs.¹²

Data-driven surrogate modeling offers a practical alternative. An artificial neural network (ANN) surrogate trained on 500 finite-element samples predicts lower-bound buckling loads for cylindrical shells under localized axial compression with 67% of test cases showing errors below 5% and 87% below 10%.¹³ Physics-informed neural networks replace repeated nonlinear simulations for axially compressed shells while retaining equilibrium consistency,¹⁴ while also being used to evaluate compression buckling of laser-welded stiffened plates.¹⁵ Further, a hybrid machine-learning framework combines generalized surrogates with interval analysis to perform uncertain buckling assessment.¹⁶ Surrogate-driven optimization of variable-stiffness laminates further demonstrates the scalability of these approaches to larger design spaces.¹⁷

In Table 1, we present a comprehensive summary of 15 recent studies leveraging machine learning (ML) techniques to predict buckling behavior in structural components. These studies span a diverse set of geometries and structural forms, including columns, plates, shells, and beams, with dataset sources ranging from finite element simulations and experimental results to prior literature. ANNs appear most frequently, reflecting their strong predictive capabilities across various contexts. Hybrid and ensemble models such as ANFIS variants, gradient boosting regression (GBR), and CatBoost also demonstrate notable success, particularly in enhancing prediction accuracy and model stability. Key test metrics reported include mean squared error (MSE), root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R²), among others. Noteworthy findings include the superior performance of ANN in predicting buckling loads for complex geometries,^18–21 the effectiveness of polynomial regression for additively manufactured panels,²² and the identification of beam geometric parameters as critical features in ensemble-based models.²³ Furthermore, CatBoost and GBR models have emerged as top performers in recent comparative studies,^24,25 suggesting the growing potential of boosting-based algorithms in structural stability prediction.

Table 1.

Comprehensive summary of recent studies utilizing ML techniques to estimate buckling load.

Study	Year	Structure	Dataset source	ML model(s)	Test metrics	Findings
Mustafa¹⁸	2024	I-shaped steel columns	Synthetic dataset from previous research	ANN, RF, KNN	R ², VAF, RMSE, MAE, MAD	ANN best for column load, superior to RF and KNN
Jayabalan et al.¹⁹	2022	Rectangular steel plates with circular opening	Prior literature	GEP, ANN, EPR	SSE, R², Average error	ANN and EPR highest accuracy with GEP lowest
Sun et al.²⁰	2021	Hat-stiffened panel	FEA simulations	ANN-Pb, ANN-Mb	Mean error, R²	ANNs predict buckling load/mode with ≥95% accuracy
Rehman Tahir et al.²¹	2021	Thin-walled cylindrical shells	Prior literature	ANN	MSE, RMSE, MAE, R²	ANN buckling loads agree with experimental buckling loads
Rayhan et al.²²	2025	Additively manufactured panels	FEA simulations	MLR, PR, RF, KNN	MAE, MSE, RMSE, R²	Polynomial regression provided the highest accuracy
Degtyarev and Tsavdaridis²³	2022	Steel cellular beams	FEA simulations	DT, RF, KNN, GBR, XGB, LightGBM, CatBoost	MSE, MAE, MAPE, R²	Optimized models excellently match FEA
Ahmed et al.²⁴	2024	C-section thin-walled composite structures	FEA simulations	LR, Lasso regression, DT, RF, GBR	R ², RMSE, MSE	GBR best for composite structure buckling
Hajdú et al.²⁵	2024	Corrugated web I-beams	FEA simulations	DT, RF, GBR, SVM, CatBoost, DNN	Multiple metrics	CatBoost method has the best performance and the algorithm has the smallest scatter
Schweizer et al.²⁶	2025	Columns, plates, cylindrical shells	FEA with Monte Carlo	ANN	MSE, RMSE	ANN predicts buckling under uncertainty
Ly et al.²⁷	2019	I-shaped cellular steel beams with circular openings	Prior literature	ANFIS-RCSA, ANFIS-CA, ANFIS-SFLA	RMSE, MAE, R	ANFIS-SFLA most accurate hybrid model
Nguyen et al.²⁸	2021	Y-shaped cross-section steel columns	Experimental results of 57 buckling tests	ANN	RMSE, MAE, R	Critical metrics for choosing ML model are RMSE and MAE
Abambres et al.²⁹	2019	I-section cellular steel beams	FEA simulations	ANN	Relative error	Max error 3.7%; average error 0.4%; only 0.1% > 3%
Zhu et al.³⁰	2021	Imperfect reticulated shell	FEA simulations	ANN, SVR	Relative error	Over 90% of samples predicted within 10% error
Lin et al.¹³	2025	Thin-walled cylindrical shells	FEA simulations	ANN	Relative error, MSE, R	Most samples presented relative errors within ±10%
Mojtabaei et al.³¹	2023	Thin-walled channel elements	FSM, ENFM	ANN	MSE, MAPE, MAE, R²	Predictions of buckling loads and modal decompositions achieve 98% accuracy

Altogether, the widespread application of ML in buckling load estimation is well justified, given its high predictive accuracy and ease of use. Whether coupled with finite element analysis (FEA) or classical stability theories, ML models offer efficient, scalable alternatives to costly numerical analyses, enabling rapid evaluation across large parameter spaces and supporting data-driven design in complex structural systems.

While ML has been widely applied to buckling prediction across a variety of structural forms, direct transfer of these models to tapered AFG nanobeams governed by nonlocal elasticity and supported by elastic foundations is not straightforward. In this regime, stiffness is modulated simultaneously by material gradation, geometric taper, and foundation interaction, with each effect further shaped by the chosen boundary conditions. These factors interact nonlinearly, producing parameter couplings absent in conventional column, plate, or shell datasets. Models trained on those simpler geometries cannot reliably extrapolate here, as nonlocal effects shift stiffness and stability thresholds in a manner that depends on both load and support conditions. Without a tailored predictive framework, designers of micro-electromechanical systems and nanoscale devices face costly simulations and limited ability to explore lightweight, stability-critical designs.

In this work, we construct a high-fidelity, synthetic dataset spanning the coupled space of gradation, taper, foundation stiffness, nonlocal parameter, and boundary condition, and use it as a controlled testbed to evaluate four supervised learning strategies: linear regression (LR), polynomial regression (PR), support vector machine (SVM), and artificial neural network (ANN). This convergence of advanced spectral solution methods for rapid data generation with modern nonlinear ML architectures enables, for the first time, tractable multi-factor buckling prediction in tapered AFG nanobeams. Compared with traditional numerical methods, which require repeatedly solving governing equations for each potential design configuration and boundary condition, the proposed learning strategies offer a highly efficient alternative. Once trained on representative datasets, these models can (i) capture complex nonlinear relationships between the design attributes and the corresponding critical buckling loads across various boundary conditions without relying on explicit differential formulations; (ii) instantly estimate buckling responses for new design configurations within the training domain, making them highly attractive for optimization and design applications while eliminating the need for repeated meshing or re-computation.

This paper is structured as follows. Section 2 details the methodological framework, including dataset construction, statistical analysis, and model implementation. Section 3 presents the results, comparing model performance across BCs using standard evaluation metrics. Section 4 concludes with key findings and recommendations for future work.

Methods

This section outlines the methodology to develop an ML framework for predicting critical buckling loads of tapered AFG nanobeams on elastic foundations. First, a dataset is generated for three boundary conditions (BCs): clamped–clamped (C–C), simply supported (S-S), clamped–simply supported (C–S) by performing a parametric sweep of beam geometry, material gradation and nonlocal parameters. Second, feature relevance and independence are ensured via statistical preprocessing using Pearson correlation coefficient (PCC) and variance inflation factor (VIF). Third, four regression models LR, PR, SVM, and ANN are trained and validated using an 80/20 hold-out strategy repeated over multiple trials. Model performance is evaluated using R², RMSE, MAE, and computational time on validation and test sets. Finally, predictive accuracy and robustness are assessed through predicted versus true comparisons and residual analysis to confirm superior generalization.

Data description

This subsection outlines the data-generation procedure used to predict critical buckling loads of tapered AFG nanobeams on elastic foundations. Figure 1 illustrates the beam geometry and the nature of the BCs under investigation. Our approach, like prior literature,^18,19,21 relies on datasets drawn from earlier works. A comprehensive description of dataset generation and the governing equation is presented here. For additional details, readers are referred to Sari et al.³² for full derivations and parameter definitions.

Figure 1.

Nonlocal AFG tapered Euler–Bernoulli beam with (a) simply supported and (b) clamped BCs resting on elastic foundation and subjected to a constant axial compressive load.

It is believed that Eringen’s nonlocal theory can be exploited for nanostructures with certain BCs as S-S and C–C. For nanobeams with S-S edges, the BCs are the deflection $w = 0$ and the bending moment $M = 0$ . Because the moment is zero at the ends and the symmetric nature of the kernel/solution in many load cases, the CBC terms reduce to zero. Therefore, the BCs force the extra constitutive terms to vanish. Moreover, for nanobeams with C–C supports, the BCs are $w = 0, w^{'} = 0$ . These kinematic constraints are strong enough to satisfy the CBC relations, and hence, no inconsistency arises.

Furthermore, there is a common belief that the nonlocal elasticity theory has a lot in common with other theories such as the strain gradient elasticity.³³ Besides, it is known that the nonlocal elasticity theory can predict the performance of a wide range of nanostructures by avoiding the most complex equations.³⁴

Additionally, the nonlocal elasticity theory remains a foundational framework that has inspired numerous refined models, including the strain-driven and stress-driven theories, as well as the nonlocal strain gradient and integral–differential hybrid theories. Therefore, despite its mathematical and physical limitations in some cases, Eringen’s nonlocal elasticity theory retains a central role in nano mechanics as a reference model and as a basis for developing more advanced theories. To ensure the reliability of Erigen’s nonlocal elasticity theory, few references in which this theory has been utilized to investigate the behavior of nanostructures have been added to the list of references, for completeness.^34–46

Hence, we adopt Eringen’s nonlocal Euler–Bernoulli beam model to derive the dimensionless governing equation of motion shown in equation (1):

\frac{d^{2}}{d X^{2}} (E (X) I (X) \frac{d^{2} W}{d X^{2}}) + K_{w} W - K_{G} \frac{d^{2} W}{d X^{2}} - μ^{2} K_{w} \frac{d^{2} W}{d X^{2}} + μ^{2} K_{G} \frac{d^{4} W}{d X^{4}} = λ (- \frac{d^{2} W}{d X^{2}} + μ^{2} \frac{d^{4} W}{d X^{4}})

(1)

Here, $X$ is the longitudinal axis of the beam, $E (X)$ is the modulus of elasticity, $I (X)$ is the area moment of inertia, $ρ (X)$ is the density of the AFG beam’s material, $A (X)$ is the beam’s cross-sectional area, $W (X)$ is transverse displacement, $λ$ is the dimensionless critical buckling load, $μ$ is the nonlocal parameter, and $K_{W}$ , $K_{G}$ the Winkler and shear stiffnesses. These parameters are expressed in equations (2) to (9) as:

X = \frac{x}{L}

(2)

W = \frac{w}{L}

(3)

μ = \frac{e_{0} a}{L}

(4)

λ = \frac{P L^{2}}{E_{0} I_{0}}

(5)

K_{W} = \frac{k_{w} L^{4}}{E_{0} I_{0}}

(6)

K_{G} = \frac{k_{G} L^{2}}{E_{0} I_{0}}

(7)

E (X) = \frac{E (x)}{E_{0}}

(8)

I (X) = \frac{I (x)}{I_{0}}

(9)

Where $L$ is the beam’s length, $I_{0}$ , $A_{0}$ , $E_{0}$ , and $ρ_{0}$ are the area moment of inertia, the cross-sectional area, the modulus of elasticity, and the density at $x = 0$ , respectively. Substituting equations (2) to (9) into equation (1) yields an eigenvalue problem whose solutions are the set of $λ$ values. The Chebyshev spectral collocation method is utilized to discretize the governing equation and boundary conditions. For structures modeled as Euler–Bernoulli beams, two boundary requirements must be fulfilled at the beam’s edges. Given that there is a single unknown at each point $W_{i}$ , the Chebyshev collocation method can be employed to satisfy only one BC. To address this challenge, the boundary conditions are implemented by articulating the displacement at the boundary point and its neighboring point in relation to the displacement at other sites inside the domain. For example, the boundary conditions of a beam with fixed endpoints are specified as:

Applying the Chebyshev collocation method, these conditions are expressed in equation (10):

D 1_{(1, 1)} W_{1} + D 1_{(1, 2)} W_{2} + D 1_{(1, N)} W_{N} + D 1_{(1, N + 1)} W_{N + 1} = - \sum_{k = 3}^{N - 1} D 1_{(1, k)} W_{k}, W_{1} = 0

(10)

Where $D 1$ is the Chebyshev differentiation matrix and the subscripts represent the matrix indices. The Chebyshev collocation points are defined on the interval (−1, 1) in equation (11) as Trefethen⁴⁷:

x_{j} = \cos (\frac{j π}{N}), j = 0, 1, \dots ., N

(11)

In the current study, the initial Chebyshev points are moved to fall within the interval of (0, 1) as the x-axis is normalized to this range. Since the elements of the Chebyshev differentiation matrices depend on the distribution of the points, they will differ from those derived by Trefethen.⁴⁷ Equation (10) is expressed in matrix vector form as equation (12):

[\begin{matrix} D 1_{(1, 2)} & D 1_{(1, N)} \\ D 1_{(N + 1, 2)} & D 1_{(N + 1, N)} \end{matrix}] {\begin{matrix} W_{2} \\ W_{N} \end{matrix}} = {\begin{matrix} - \sum_{k = 3}^{N - 1} D 1_{(1, k)} W_{k} \\ - \sum_{k = 3}^{N - 1} D 1_{(N + 1, k)} W_{k} \end{matrix}}

(12)

Considering equation (12), $W_{2}$ and $W_{N}$ are written in equation (13) as:

\begin{matrix} W_{2} = \frac{1}{Det} \\ (- D 1_{(N + 1, N)} \sum_{k = 3}^{N - 1} D 1_{(1, k)} W_{k} + D 1_{(1, N)} \sum_{k = 3}^{N - 1} D 1_{(N + 1, k)} W_{k}), \\ W_{N} = \frac{1}{Det} \\ (D 1_{(N + 1, 2)} \sum_{k = 3}^{N - 1} D 1_{(1, k)} W_{k} - D 1_{(1, 2)} \sum_{k = 3}^{N - 1} D 1_{(N + 1, k)} W_{k}) \end{matrix}

(13)

where $Det$ is the determinant of the 2 × 2 matrix in the left-hand side of equation (13) and is given by equation (14):

Det = (D 1_{(1, 2)} \times D 1_{(N + 1, N)}) - (D 1_{(1, N)} \times D 1_{(N + 1, 2)})

(14)

Thus, the Chebyshev collocation matrices are adjusted as in equation (15):

\bar{Dn} (i, k) = \sum_{k = 3}^{N - 1} D n_{(i, k)} + \frac{D n_{(i, 2)}}{Det} (- D 1_{(N + 1, N)} \sum_{k = 3}^{N - 1} D 1_{(1, k)} + D 1_{(1, N)} \sum_{k = 3}^{N - 1} D 1_{(N + 1, k)}) + \frac{D n_{(i, N)}}{Det} (D 1_{(N + 1, 2)} \sum_{k = 3}^{N - 1} D 1_{(1, k)} - D 1_{(1, 2)} \sum_{k = 3}^{N - 1} D 1_{(N + 1, k)})

(15)

where $i = 3, 4, \dots, N - 1$ , and $Dn = D 1^{n}$ . Equation (16) is discretized by the Chebyshev collocation method as shown in equation (16):

(\begin{matrix} (R 1 \times \bar{D 2}) + (R 2 \times \bar{D 3}) + (R 3 \times \bar{D 4}) + \\ K_{w} I - K_{G} \bar{D 2} - μ^{2} K_{w} \bar{D 2} + μ^{2} K_{G} \bar{D 4} \end{matrix}) {W} = λ (- \bar{D 2} + μ^{2} \bar{D 4}) {W}

(16)

where $I$ is a $(M - 4) \times (M - 4)$ identity matrix, $R 1$ , $R 2$ , and R3 are diagonal $(M - 4) \times (M - 4)$ matrices with values of $R 1_{i}$ , $R 2_{i}$ , and $R 3_{i}$ on the main diagonal, respectively, expressed as (equation (17)):

\begin{matrix} R 1_{i} = (2 c_{b} {(c_{h} x_{i} - 1)}^{3}) + (6 c_{h} (c_{b} + 2 c_{b} x_{i} - 1) {(c_{h} x_{i} - 1)}^{2}) \\ + (6 c_{h}^{2} (1 + x_{i}) (1 - c_{b} x_{i}) (1 - c_{h} x_{i})) \\ R 2_{i} = (2 (c_{b} + 2 c_{b} x_{i} - 1) {(c_{h} x_{i} - 1)}^{3}) \\ + (6 c_{h} (1 + x_{i}) (1 - c_{b} x_{i}) (1 - c_{h} x_{i}) (c_{h} x_{i} - 1)) \\ R 3_{i} = (1 + x_{i}) (1 - c_{b} x_{i}) (1 - c_{h} x_{i}) {(c_{h} x_{i} - 1)}^{2} \end{matrix}

(17)

where $i = 1, 2, \dots, M - 4$ , and $M = N + 1$ . For simply supported ends, the boundary conditions are expressed as (equation (18)):

W (0) = \frac{d^{2} W (0)}{d X^{2}} = W (1) = \frac{d^{2} W (1)}{d X^{2}} = 0

(18)

The boundary conditions are expressed using the spectral collocation method as in equations (15) and (16), except that each differentiation matrix $D 1$ is substituted with the differentiation matrix $D 2$ . Moreover, for beams that are clamped at $x = 0$ and simply supported at $x = 1$ , the boundary conditions are given as (equation (19)):

W (0) = \frac{dW (0)}{dX} = W (1) = \frac{d^{2} W (1)}{d X^{2}} = 0

(19)

The spectral collocation method can be used to discretize these conditions as explained earlier. Once the governing equation and the boundary conditions are discretized, the inputs (hereafter denoted as $x_{i}, i = 1, \dots, 5$ ) to this model are introduced as:

$x_{1}$ . A dimensionless nonlocal parameter related to material or geometry, $μ$ .

$x_{2}$ and $x_{3}$ . Normalized shape parameters, $c_{b}$ and $c_{h}$ .

$x_{4}$ and $x_{5}$ . Stiffness-related coefficients, $K_{W}$ and $K_{G}$ .

The typical ranges of these inputs are (0, 1), (0, 0.8), (0, 0.8), (0, 400), and (0, 10), for $x_{1}$ through $x_{5}$ , respectively.³²

Proposed methodology

This section presents the methodology proposed in this work to estimate the critical buckling loads of three different BCs, as depicted in Figure 2. Specifically, let $X$ and $Y$ be the input and output matrices, containing the design parameters and the associated critical buckling loads, respectively.

Figure 2.

The methodology proposed to estimate the critical buckling loads for three different BCs of a tapered nanobeam under study.

The proposed modeling methodology comprises the following four systematic steps:

Step 1. Dataset generation: This step entails establishing two different datasets representing the tapered nanobeam under study for three different BCs ( $B C_{j}, j = 1, 2, 3$ ), covering the CC, SS, and CS cases. Specifically, the training dataset (hereafter denoted by $D_{Train}$ ) is constructed to develop data-driven predictive models, with inputs sampled uniformly across the feasible design space (hereafter denoted by $X_{Train}$ ), using a step size $R_{1}$ . The output includes the corresponding three critical buckling loads, $P_{CC}$ , $P_{SS}$ , and $P_{CS}$ (hereafter denoted by $Y_{Train}$ ). A separate test dataset (hereafter denoted by $D_{Test}$ ) is independently generated using a coarser and offset grid with a different step size $R_{2}$ , such that the test points lie strictly between the training grid points. This ensures that all test inputs ( $X_{Test}$ ) and their corresponding outputs ( $Y_{Test}$ ) are completely unseen during model training and validation and used to assess the generalization capability of the trained models.

It is worth mentioning that even though the Chebyshev spectral collocation method (Section 2.1) has been proven effective in providing highly accurate and numerically exact solutions to the governing equations at hand, the established datasets ( $D_{Train}$ and $D_{Test}$ ) may not fully represent all the inherent complexities and stochastic imperfections present in real nanoscale structures. Nevertheless, the datasets have been carefully designed to cover a representative range of input parameters, thereby allowing the ML models to be developed in later steps (i.e. Step 3) to generalize effectively to unseen or slightly perturbed conditions. In practice, such a balance between mathematical precision and parametric diversity enables the developed ML models to serve as robust predictors within the physically meaningful design space.

Step 2. Statistical analysis: This step involves computing the statistical association between each input parameter and each corresponding output of the training dataset ( $D_{Train}$ ) to better understand the relationships and ensure the effectiveness of the model development process of Step 3. Specifically, the PCC⁴⁸ is computed to quantify the linear correlation between each input parameter and each of the three critical buckling loads. Indeed, higher absolute values of PCC indicate stronger linear dependence and vice versa.

Moreover, to detect and mitigate multicollinearity among the input parameters, the VIF⁴⁹ is computed. In practice, multicollinearity leads to unstable and unreliable regression models. Typically, as a rule of thumb, VIF > 5 indicates problematic levels of collinearity and design parameters with high VIF values should then be removed or transformed.

In summary, this step has double effects by ensuring that the selected features are both (i) relevant (through the computation of PCC) and (ii) independent (through the computation of VIF), supporting a stable and interpretable ML model development process of Step 3.

Step 3. ML development: Once the datasets are established and the input-output parameters are statistically analyzed, the refined dataset is utilized for ML model development. In parallel, the test dataset generated in Step 1 is reserved for the model evaluation process in Step 4. In this step, several supervised regression techniques are implemented and compared to evaluate their suitability in estimating the critical buckling loads for the considered BCs. For illustrative and benchmarking purposes, various models are explored, ranging from simple linear to highly nonlinear models, as follows:

• Linear regression (LR): The LR assumes a simple linear relationship between the inputs and each output, individually. The model development stage aims to estimate the $β$ coefficients (the y-intercept $β_{o}$ and the slopes for each $i$ th input $β_{i}, i = 1, \dots, 5$ ) to fit the linear equation (equation (20)):

\hat{P} = β_{o} + \sum_{i = 1}^{5} β_{i} x_{i}

(20)

The objective is to minimize the residual sum of squares (RSS) between the individually estimated ( $\hat{P}$ ) and actual ( $P$ ) critical buckling loads of each BC. Independent LR models are developed for each boundary condition, $B C_{j}, j = 1, 2, 3$ . The schematic representation of the LR model is shown in Figure 3.

Figure 3.

LR schematic.

• Polynomial regression (PR): The PR extends the LR to include second-degree terms and cross-products, allowing the model to capture basic curvature in the training dataset. The objective is then to find the optimal quadratic equation that minimizes the mismatch between the individually estimated ( $\hat{P}$ ) and actual ( $P$ ) critical buckling loads of each BC (equation (21)):

\hat{P} = β_{o} + \sum_{i = 1}^{5} β_{i} x_{i} + \sum_{i = 1}^{5} β_{ii} x_{i}^{2} + \sum_{i < j} β_{ij} x_{i} x_{j}

(21)

This allows more flexibility in modeling the relationship without introducing higher model complexity as in the subsequent models investigated. Independent PR models are developed for each boundary condition, $B C_{j}, j = 1, 2, 3$ . The schematic representation of the PR model is shown in Figure 4.

Support vector machines (SVMs): The SVM model uses a nonlinear kernel function ( $K (.)$ ) (i.e. for similarity mapping) to transform the input space ( $X_{Train}$ ) into a high-dimensional feature space, where it identifies a hyperplane that best fits the data within a predefined error margin. Specifically, the SVM model relies on a subset of the training samples, known as support vectors, which define this hyperplane. Default MATLAB^® settings are used for kernel type, box constraints, and epsilon margin, for clarification purposes. Independent SVM models are developed for each boundary condition, $B C_{j}, j = 1, 2, 3$ . The schematic representation of the SVM structure is shown in Figure 5.

Artificial neural networks (ANNs): The ANN regression model is adopted to approximate complex nonlinear mathematical relationships between the design input parameters and each critical buckling load response in output. The model learns from a set of historical/representative data (i.e. the training dataset, $D_{Train}$ ) and accurately estimates/predicts the corresponding output values for new, unseen inputs (i.e. validation or test datasets). The ANN is developed using default MATLAB^® architecture and training settings, which typically involve a single hidden layer with a predefined number of neurons (set by a trial-and-error procedure, $h = 1, \dots, H$ ; the final value of $H$ is 10 for all boundary conditions), a sigmoid activation function for the hidden layer ( $f_{1} (.)$ ), a linear activation function for the output layer ( $f_{2} (.)$ ) and the back-propagation (BP) Levenberg–Marquardt (LM) learning/training algorithm, which iteratively minimizes the error or mismatch between the actual ( $P$ ) and predicted ( $\hat{P}$ ) outputs. Nevertheless, other BP algorithms such as Bayesian regularization (BR) and scaled conjugate gradient (SCG) can also be utilized for this regard. In practice, the LM algorithm is a compromise between BR and SCG in terms of computational effort and prediction accuracy, and it has been shown effective in nonlinear regression problems across various engineering applications. Independent ANN models are developed for each boundary condition, $B C_{j}, j = 1, 2, 3$ . The schematic representation of the ANN structure is shown in Figure 6.

To ensure robust and generalizable results, all models are trained and validated using a 20% hold-out validation strategy, while repeating the simulation $R$ different times for robustness in the ultimate performance outcomes. This process helps reduce the risk of overfitting and provides a more reliable estimation of each model’s expected performance across different subsets of the input space. Specifically, 20% of the $D_{Train}$ are held out for validation, while the remaining 80% of the $D_{Train}$ are kept for training the predictive models.

Figure 4.

PR schematic.

Figure 5.

SVM schematic.

Figure 6.

ANN schematic.

Model evaluation

The model’s performance is quantified using standard regression evaluation metrics, including:

RMSE (equation (22)). It measures the average magnitude of estimation/prediction mismatch. Small values reflect the effectiveness of the built-model and vice versa.

RMSE = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {({\hat{P}}_{i} - P_{i})}^{2}}

(22)

MAE (equation (23)). It measures the average absolute mismatches between estimated/predicted and actual buckling load values. Small values reflect the effectiveness of the built-model and vice versa.

MAE = \frac{1}{N} \sum_{i = 1}^{N} | {\hat{P}}_{i} - P_{i} |

(23)

R ² (equation (24)). It reflects the proportion of variance in the output explained by each built-model. Higher values reflect the effectiveness of the built-model and vice versa.

R^{2} = 1 - \frac{\sum_{i = 1}^{N} {({\hat{P}}_{i} - P_{i})}^{2}}{\sum_{i = 1}^{N} {(P_{i} - \bar{P})}^{2}}

(24)

where $P_{i}$ and ${\hat{P}}_{i}$ are the actual and predicted $i$ th buckling load, respectively, $\bar{P}$ is the mean of the actual values, and $N$ is the total number of observations.

In addition to the above-mentioned performance metrics, the computational time (Time) required to train each model is also recorded during its development to provide insight into the trade-off between accuracy and efficiency.

The hold-out validation process is repeated $R$ times, and the performance metrics are recorded for each trial. After completing the $R$ simulation trials, the average performance metrics and their standard deviations are computed to quantify the variability resulting from different data splits. Once the models are trained using the respective training portions, their generalization performance is evaluated using the independent test dataset generated in Step 1 ( $D_{Test}$ ). This dataset contains input configurations that were not seen during training and thus provides a reliable estimate of each model’s predictive capability in real-world scenarios.

Step 4. Insights and recommendations: Once the models are built and evaluated for each BC of the beam under study, insights and recommendations, from a practical point of view, are highlighted.

Results and discussion

This section presents the results obtained by the application of the proposed methodology, step by step.

The training ( $D_{Train}$ ) and test ( $D_{Test}$ ) datasets are established, using the respective ranges of the input design parameters, sampled uniformly with $R_{1} = 10$ and $R_{2} = 5$ potential points in their respective ranges, respectively. The established datasets have the sizes of 100,000 × 8 and 3125 × 8, respectively, comprising five design parameters in input and three corresponding critical buckling loads in output representative of the three BCs under study.

Figure 7 presents the PCC values obtained for each input design parameter with respect to each buckling load of the three BCs examined, that is, $P_{CC}$ , $P_{SS}$ , and $P_{CS}$ . The results indicate consistent trends across all three buckling loads, with $μ$ (which should be minimized to enhance buckling performance) showing the strongest negative correlation, that is, −0.6089, −0.5856, and −0.6000 for the three buckling loads, respectively, and $K_{G}$ (which should be maximized to enhance buckling performance) showing the strongest positive correlation, that is, 0.2317, 0.3454, and 0.3169 for the three buckling loads, respectively. Furthermore, the computed VIF values for all five design parameters are equal to 1, confirming the absence of multicollinearity in the training dataset.

Figure 7.

PCC values obtained for the five design parameters with respect to each critical buckling load.

The training dataset is then fed to the four selected ML models to estimate the associated buckling loads, evaluated individually on both the validation and test data points across $R = 10$ independent simulation trials. Specifically, Figure 8 presents the average performance metrics (shown as bars), and their corresponding standard deviations ( $σ$ , shown as error bars) computed over the 10 replicates for each of the four models on the validation set. From the figure, the following insights can be drawn:

The ANN model clearly outperforms the other models across all performance metrics, achieving RMSE values below 1.1 (in normalized units), MAE values below 0.67 (in normalized units), and R² values exceeding 0.99 for all three BCs.

The ANN performance is also shown to be consistent across the three BCs, in contrast to the larger variability observed in the performance of the other models. This highlights the ANN’ robustness and its ability to accurately estimate the buckling loads across different structural configurations.

The relatively poor performance of the LR model suggests a highly nonlinear relationship between the input design parameters and the output buckling loads. This is further supported by the improved performance of the PR model. Conversely, the SVM model performed worse than PR and LR in terms of predictive accuracy, implying that either extensive hyperparameter tuning is needed or that alternative modeling techniques such as ANN are more suitable for this application.

The SVM and ANN models exhibit slightly higher variability across the 10 runs compared to the LR and PR models. This is expected, as both SVM and ANN involve hyperparameter initialization and iterative training (i.e. model uncertainty), which introduce additional sources of uncertainty beyond data splitting. Specifically, the standard deviations ( $σ$ ) of the performance metrics for the LR, PR, SVM, and ANN models across the three boundary conditions range between ∼0.0258–0.0611, 0.0309–0.0557, 0.0555–0.1075, and 0.1124–0.3635 for RMSE, 0.0124–0.0224, 0.0165–0.0330, 0.0210–0.2830, and 0.0707–0.2654 for MAE, and 0.0029–0.0038, 0.0022–0.0027, 0.0041–0.010, and 0.0012–0.0104 for R², respectively.

Table 2 reports the computational efforts required by the models for their development across the 10 replicates for each boundary condition. As expected, the LR required the least computational effort, while the ANN model exhibited the highest. Moreover, a noticeable variability was observed across the 10 runs, particularly for the ANN model, reflecting sensitivity to initialization and training dynamics.

Figure 8.

Performance metrics obtained by each of the four models on the validation set across the 10 trials, together with the corresponding standard deviation ( $σ$ ) values: (a) RMSE, (b) MAE, and (c) R².

Table 2.

Computation time in seconds required by the predictive models on the validation sets across the 10 simulation trials.

Model	C–C (s)	S-S (s)	C–S (s)
LR	0.0351 ± 0.0051	0.0365 ± 0.0052	0.0400 ± 0.0090
PR	0.0604 ± 0.0082	0.0653 ± 0.0156	0.0634 ± 0.0110
SVM	61.2890 ± 4.3969	975.5923 ± 44.2587	112.3869 ± 13.2894
ANN	28.7416 ± 7.8801	31.8245 ± 3.1607	32.2178 ± 4.6079

Similar insights are observed when the models are applied to the unseen test dataset across the 10 runs, as depicted in Figure 9. The ANN model maintains its superior performance across all three metrics and BCs, demonstrating strong generalizability and robust predictive capability on unseen data. For clarity, the standard deviations ( $σ$ ) of the performance metrics for the LR, PR, SVM, and ANN models across the three boundary conditions on the test dataset range between ∼0.0023–0.0025, 0.0011–0.0031, 0.0192–0.0528, and 0.1061–0.3140 for RMSE, 0.0066–0.0085, 0.0039–0.0056, 0.0032–0.3133, and 0.0982–0.2746 for MAE, and 0.00026–0.00033, 0.0001–0.00025, 0.0023–0.0086, and 0.0016–0.0090 for R², respectively.

Figure 9.

Performance metrics obtained by each of the four models on the test dataset across the 10 trials, together with the corresponding standard deviation ( $σ$ ) values: (a) RMSE, (b) MAE, and (c) R².

To further illustrate the models’ capability in estimating the three buckling loads, Figure 10 shows the predicted versus true/actual load values for each model and BC on the validation set, shown for one arbitrary replicate (i.e. replicate #1). The ANN consistently demonstrates the best alignment with the ideal diagonal line (dashed line) across all conditions, confirming its superior predictive accuracy and generalization. In contrast, the LR and PR models show systematic underestimation, especially at higher load values. The SVM model significantly deviates from the ideal behavior, indicating limited predictive capability. These patterns reinforce the superior performance of the ANN model, as previously highlighted by the quantitative metrics.

Figure 10.

Predictive versus true/actual buckling load values for the four models across the three BCs: (a) C–C, (b) S-S, and (c) C–S.

For illustration purposes, Figure 11 presents the true (actual) and predicted load values for one BC, specifically, the C–C case, using the ANN model on the test dataset for the first 50 examples (Figure 11(a)) and the associated prediction residuals (mismatch; Figure 11(b)). The predicted values are shown to consistently follow the trend of the actual loads, indicating the model’s capability to capture the underlying patterns and variations in the data.

Figure 11.

(a) True versus predicted buckling load values for the C–C case using the ANN model on the first 50 test data points and (b) the associated prediction residuals (mismatch).

Buckling is a critical factor in structural and mechanical engineering as it can cause abrupt and catastrophic failure with minimal warning, unlike gradual failures such as yielding. It is a stability problem in which a structure loses its load-carrying capacity due to geometric deformation, even though the material itself remains intact. This is especially substantial in slender components like columns, beams, plates, and shells that are widely used in bridges, towers, buildings, aircraft, and vehicles where lightweight and material efficiency are prioritized. These elements are highly susceptible to instability under compressive loads, making accurate prediction of critical buckling loads essential to avoid failure and improve designs. In high-performance applications such as aerospace and automotive industries, the use of thin-walled structures for weight reduction further increases vulnerability to buckling, where even small imperfections or load variations can trigger instability. As a result, buckling analysis is critical for selecting appropriate shapes, materials, and safety features, as well as informing inspection methods, maintenance planning, and design code formulation to guarantee structures are safe, efficient, and long-lasting.

Conclusions and future directions

This paper developed an ML framework to predict the critical buckling loads of tapered AFG nanobeams on elastic foundations using non-local elasticity theory. A synthetic dataset was generated through a comprehensive parametric sweep of design inputs across three boundary conditions (C–C, S-S, C–S). Feature screening was performed using PCC and VIF to ensure input relevance and independence. Four regression models (LR, PR, SVM, ANN) were trained and evaluated using repeated hold-out validation and standard error metrics on both validation and test sets. Key findings include:

PCC analysis revealed that $μ$ had the strongest negative influence on buckling load (PCC ≈ −0.61), while $K_{G}$ had the strongest positive (PCC ≈ 0.35), providing interpretable physical insight.

All design inputs passed the VIF threshold (VIF = 1), confirming no multicollinearity and supporting model stability.

ANN achieved R² > 0.99, RMSE < 1.1 (in normalized units), and MAE < 0.67 (in normalized units) across all BCs on both validation and test datasets, indicating high predictive accuracy and generalization.

The ANN model showed strong consistency across different BCs, with low performance variance, making it robust to structural configuration changes.

LR failed to capture the nonlinear input-output relationships, with significantly lower R² and higher error metrics.

PR improved upon LR, but both models still underestimated loads, especially at higher values, and showed limited generalization to unseen data. This contrasts with previous findings, such as Rayhan et al.,²² where the PR model, optimized with respect to the polynomial degree, achieved the highest accuracy. In the present study, a second-degree PR model was adopted to illustrate the transition from a linear model (LR), to nonlinear (PR), and highly nonlinear (SVM and ANN) models, while assessing their predictive capabilities across the considered metrics. Therefore, future studies in this domain could focus on evaluating the influence of higher-degree terms on prediction accuracy, as well as comparing additional diverse models, since performance appears to be case-dependent.

SVM showed poor accuracy and high variance in predictions, likely due to default kernel settings and sensitivity to hyperparameters.

ANN predictions closely tracked true buckling loads across all 50 test examples in the C–C case, with low residuals and minimal bias.

ANN required moderate training time (∼28–32 s) and offered the best trade-off between accuracy, robustness, and efficiency.

SVM incurred the highest computational cost (up to 975.6 s in S-S), with no advantage in prediction quality.

While the ANN model provided excellent predictive performance, several areas remain open for further development. The current study relies on a synthetic dataset based on idealized formulations and extending the approach to experimental or real-world datasets would help validate applicability by capturing potential imperfections inherent in nanoscale structures, including fabrication tolerances, surface defects, and thermal or geometric variations. In addition, while ANN proved reliable, incorporating uncertainty quantification or probabilistic bounds, through techniques such as Monte Carlo simulations, would enhance confidence in safety-critical settings. Future work could also explore transfer learning, physics-informed networks, or hybrid ML-FEA pipelines to support broader generalization and interpretability across different nano structural forms and loading scenarios.

Footnotes

Handling Editor: Aarthy Esakkiappan

ORCID iDs

Sameer Al-Dahidi

Ma’en S. Sari

Author contributions

Sameer Al-Dahidi: conceptualization, methodology, software, formal analysis, validation, investigation, writing – original draft, writing – review and editing, visualization. Ma’en S. Sari: conceptualization, methodology, data curation, software, writing – original draft, visualization, formal analysis.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

Data will be made available on request.

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Critical buckling load prediction in tapered axially functionally graded nanobeams using machine learning under different boundary conditions

Abstract

Keywords

Introduction

Methods

Data description

Proposed methodology

Model evaluation

Results and discussion

Conclusions and future directions

Footnotes

ORCID iDs

Author contributions

Funding

Declaration of conflicting interests

Data availability statement

References