A Hybrid Gaussian Process-Extreme Gradient Boosting Machine Learning Framework for Robust Knockdown Factor Prediction and Reliability-Based Design of Thin Spherical Shell Caps

Budhaditya De; Rohan Majumder; Devasmit Dutta; Aman Deep Gupta; Sudib Kumar Mishra

cs.LGOct 11, 2026

A Hybrid Gaussian Process-Extreme Gradient Boosting Machine Learning Framework for Robust Knockdown Factor Prediction and Reliability-Based Design of Thin Spherical Shell Caps

Authors: Budhaditya De, Rohan Majumder, Devasmit Dutta, Aman Deep Gupta, Sudib Kumar Mishra

Organizations: Department of Civil and Environmental Engineering, University of California, Los Angeles, CA, USA · Faculty of Engineering Science and Technology, Adani University, Ahmedabad, Gujarat, India · Department of Civil Engineering, Indian Institute of Technology Kanpur, Uttar Pradesh, India

Abstract

Elastic buckling remains a critical concern in the design of deep spherical shell caps, given that experiments reveal a significant reduction of critical loads from those predicted by linearized eigen-buckling analysis. This discrepancy has been attributed to initial geometric imperfections and is often reconciled through a dimensionless parameter, namely the knockdown factor (KDF). However, conventional KDFs (NASA SP-8007; Eurocode 3) are highly conservative, as they represent the lower bound of an extensive body of experimental data. This study aims to narrow the conservatism in the KDFs for spherical shell caps through machine learning (ML) regression models trained on experimental data on spherical shell buckling. A hybrid model is proposed by combining Gaussian Process Regression (GPR) with Extreme Gradient Boosting (XGBoost), and its improved performance is demonstrated over conventional ML algorithms, i.e., the Artificial Neural Network (ANN), Support Vector Regressor (SVR), Random Forest Regressor (RFR), and Histogram-based Gradient Boosting (HGBoost). Each of these models uses six input features and a single output, i.e., the KDF. The most critical feature affecting the KDF is ascertained through a sensitivity analysis. The study concludes with a reliability-based design framework that relies on the predictive ML model to obtain safety-consistent, yet less conservative, KDFs.

Figures & tables

Figure 1: Spherical shell cap (adapted from [ 41 ] ) subjected to uniform external pressure.

Figure 2: Spherical shell: (a) geometry; (b) uniform external pressure with clamped boundary condition; and (c) FE mesh [ 41 ] .

Parameter	Minimum	Maximum	Mean	Std. deviation
$R/t$	10.031	4345.745	778.519	651.191
$H/r_{0}$	0.023	1.000	0.246	0.283
$\lambda$	1.581	140.107	18.482	24.895
$Y/E$	0.001	0.0096	0.003052	0.002806
$\nu$	0.277	0.410	0.3322	0.0379
KDF	0.100	1.089	0.562	0.207

Table 1: Statistical summary of the data used to develop the ML predictive models.

Figure 3: (a) Schematic of the ANN predictive model; sensitivity of (b) $R^{2}$ and (c) RMSE to the number of hidden-layer neurons; and (d) ANN model loss versus number of epochs for the training and validation sets.

Figure 4: Performance of the ANN model on the test data with 95% confidence intervals.

Figure 5: (a) Workflow of the SVR model; (b) performance of the SVR predictive model on the test dataset with 95% confidence intervals.

Figure 6: (a) Workflow of the RFR model; (b) performance of the RFR predictive model on the test dataset with 95% confidence intervals.

Figure 7: (a) HGBoost model workflow; (b) performance of the HGBoost predictive model on the test data with 95% confidence intervals.

Figure 8: Flow diagram of the hybrid GPR–XGB model for improved KDF prediction.

Figure 9: Prediction performance of the hybrid GPR–XGB model on the test dataset with 95% confidence intervals.

Model	$R^{2}$	RMSE	MAE	95% CI coverage (%)
ANN	0.853	0.078	0.062	97.5
SVR	0.810	0.087	0.069	30.8
RFR	0.906	0.062	0.044	82.1
HGBoost	0.893	0.065	0.044	68.4
GPR–XGB	0.930	0.056	0.042	96.7
Kriging-only	0.890	0.062	0.045	95.2

Table 2: Performance of the ML predictive models on the held-out test set.

Figure 10: (a) GPR–XGB-predicted and code-stipulated KDFs versus actual KDFs; (b) GPR–XGB-predicted KDFs versus $\lambda$ .

Figure 11: Calibration curves showing nominal versus empirical confidence interval coverage (10%–95%) for the KDFs predicted by all ML models.

Figure 12: Mean absolute KDF residuals versus ML-predicted 95% confidence half-widths in different uncertainty bands.

Figure 13: KDF prediction-error distribution stratified by local data sparsity for (a) the GPR–XGB model and (b) the RFR model.

Figure 14: Sensitivity analysis of the hybrid GPR–XGB model: (a) SHAP bee-swarm plot; (b) mean absolute SHAP values; and SHAP dependence of (c) $R/t$ and (d) $H/r_{0}$ , coloured by fabrication quality $Q$ .

Figure 15: Variation of the predicted KDF with $\lambda$ for the GPR–XGB hybrid and ANN metamodels, showing the mean ( $\mu$ ) prediction and the corresponding $\pm 1\sigma$ uncertainty bounds.

Model	Mean ( $\mu$ )	$\mu+\sigma$	$\mu-\sigma$	Mean bandwidth ( $\pm 1\sigma$ )
ANN	$0.543+2.714/\lambda^{2}$	$0.557+2.790/\lambda^{2}$	$0.528+2.636/\lambda^{2}$	0.30
GPR–XGB	$0.524+3.800/\lambda^{2}$	$0.536+3.846/\lambda^{2}$	$0.512+3.754/\lambda^{2}$	0.23

Table 3: Fitted inverse-square response functions and corresponding mean prediction uncertainty bandwidths (

\pm 1\sigma

) for the ANN and GPR–XGB metamodels.

GPR–XGB
$\bar{\beta}$	$Q$ (Class B)	$\mathrm{KDF}(\mu)$	$\mathrm{KDF}(\mu+\sigma)$	$\mathrm{KDF}(\mu-\sigma)$
1	25	0.60	0.61	0.59
2		0.63	0.65	0.61
3		0.68	0.70	0.66
4		0.71	0.73	0.69
5		0.75	0.77	0.69

Table 4: Safety-consistent KDFs (mean

\mu

and

\pm 1\sigma

interval bounds) for varying target reliability (

\beta=\bar{\beta}

) for the Al-6061-T6 spherical shell cap (

R/t=400

H/r_{0}=0.58

Q=25

, Class B), obtained with the hybrid GPR–XGB and ANN metamodels.

Figure 16: Comparison of GPR–XGB-based safety-consistent design KDFs with other models for varying $\bar{\beta}$ for the Al-6061-T6 spherical shell cap ( $R/t=400$ , $H/r_{0}=0.58$ , $Q=25$ ).

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure A.1: (a)–(f) First six linearized buckling modes from eigen-analysis [ 41 ] .

Figure A.2: (a) Equilibrium load–displacement path of the perfect shell cap under uniform external pressure and (b) the pertinent LPFs at various instants of deformation [ 41 ] .

Structural Mechanics

Structural mechanics studies how structures deform, carry loads, vibrate, buckle, and fail under different materials, geometries, and conditions.

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A Physics-informed Neural Network Approach for Robust Buckling Load Prediction and Reliability-Based Design of Thin Truncated Conical Shells

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Structural Mechanics Neural Surrogate Modeling Physics-Informed ML Uncertainty Quantification

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Thin-walled truncated conical shells are widely used in aerospace, marine, offshore, and lightweight infrastructure systems due to their high strength-to-weight ratio and geometric efficiency. Their buckling resistance under axial compression, however, is highly sensitive to geometric imperfections, manufacturing tolerances, material variability, and nonlinear instability effects. Conventional design procedures rely on conservative knockdown factors (KDFs), such as those recommended in NASA SP-8019, which do not explicitly account for shell geometry, fabrication quality, data uncertainty, or target reliability. This study develops a physics-informed neural network (PiNN) framework for predicting critical buckling loads of thin truncated conical shells and integrates the trained surrogate within a reliability-based design (RBD) formulation. The model combines geometric and material descriptors with mechanics-informed features derived from shell stability theory and the localized reduced stiffness method (LRSM). A physics-informed loss function penalizes mechanically inadmissible predictions exceeding the theoretical elastic buckling load. The framework is trained and evaluated using 133 experimental Mylar conical shell tests under axial compression. Compared with a conventional deep neural network (DNN), the PiNN improves predictive accuracy, reduces mean absolute error, and enhances physical consistency. The trained PiNN is then used to evaluate reliability indices and calibrate safety-consistent KDFs for prescribed target reliability levels. Results demonstrate that the PiNN-RBD framework provides an efficient approach for uncertainty-aware design of imperfection-sensitive shell structures.

Abstract

Structural Mechanics

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