PSI - Issue 81

Oleh Yasniy et al. / Procedia Structural Integrity 81 (2026) 116–122

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criterion reflects the average deviation of predicted values from actual values as a percentage, which makes it convenient to interpret regardless of the scale of the input data. A low MAPE value indicates high model accuracy and good agreement with experimental results. The mean absolute percentage error was chosen as the forecasting error, which was calculated using the formula: = 1 ∑ | ( )− . ( )| | ( )| = 1 ∙ 100% (1) 3. Results and discussion Experimental data by Lu et al. (2019) describing the dependence of fatigue crack length a on the number of loading cycles N and the overloading factor for stress ratio R = 0.1 was used to train and test the models. Fig. 1 shows a comparison of the predicted and experimental values of fatigue crack length using the boosted trees and random forest methods. Both graphs show a high density of points along the bisector of the first coordinate angle, indicating a good match between the simulation results and the experimental data. This confirms the ability of both algorithms to accurately reproduce the physical relationship between the number of loading cycles, the overloading factor, and crack growth. The random forest model demonstrated slightly better consistency, its results are characterised by less dispersion and a more uniform fit to the ideal line. This distribution of points confirms the high predictive accuracy of the selected ensemble methods in fatigue failure modelling tasks. a b

Fig. 1. The predicted and experimental values of the crack length, obtained by method of a) random forest and b) boosted trees.

Fig. 2 shows the dependence of the fatigue crack length a on the number of loading cycles N using the random forest method. The model curves reproduce the experimental data well in all three modes, preserving the characteristic shape of the crack growth curve. The consistency is particularly noticeable at R ol = 2.0, where there is a pronounced effect of crack growth retardation after overload. The results confirm the effectiveness of the selected machine learning model in simulating complex nonlinear material behaviour under variable loading conditions. Fig. 3 shows the dynamics of the root mean square error depending on the number of trees in the models of random forest and boosted trees. In the random forest model (a), there is a decrease in error at the initial stage and its stabilisation after about 300 trees, which indicates the reliability and generalising ability of the model. In the case of boosted trees (b), the minimum error value is achieved with about 2550 trees, after which the accuracy does not improve significantly. The slight difference between the training and test data curves in both models indicates the absence of overtraining. Fig. 4 shows the histograms of the residual distribution for the models of random forest and boosted trees. For random forest, there is a high symmetry and clustering of residuals within the interval [ – 0.5; 0.5], which indicates stable model performance and low error variability. In the case of boosted trees, the distribution of residuals is wider, with several outliers in the left tail, but the bulk of the values remain close to zero. This indicates a slightly higher sensitivity of the boosted trees model to individual points, but the overall prediction accuracy remains high.