PSI - Issue 81

Dmytro Tymoshchuk et al. / Procedia Structural Integrity 81 (2026) 35–40

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The constructed dataset captured the nonlinear hysteretic nature of SMA behavior, enabling the development, training, and validation of machine learning models that predict material strain under repeated cyclic loading at different frequencies. In this study, experimentally obtained data for 100 – 250 loading – unloading cycles of the SMA material were used. The number of data samples for each frequency is presented in Table 1.

Table 1. Number of data samples in the dataset for different loading frequencies Frequency, Hz 0.3 0.5 3

5

Number of samples 14949 For the development and evaluation of the machine learning models, the constructed dataset was divided into two parts: a training set (80%) and a test set (20%). The training set was used for model training and parameter tuning, while the test set remained completely independent of the training process and was used solely for assessing the model’s performance on unseen data. 2.2. Methodology of Voting Model Construction In this study, an ensemble machine learning model of the VotingRegressor type (scikit-learn, 2025a) was employed, combining several base algorithms. The ensemble consisted of the following regressors: Random Forest Regressor (scikit-learn, 2025b), Gradient Boosting Regressor (Clark and Lee, 2025), Extra Trees Regressor (scikit-learn, 2025c), Support Vector Regressor (scikit-learn, 2025d), K-Nearest Neighbors Regressor (IBM, 2025), and a Multilayer Perceptron (MLP Regressor) (Haykin, 2009). To enhance prediction quality, weighted voting was applied. The weights of each base algorithm were determined as the inverse values of the mean squared error (MSE). This ensemble construction methodology enabled a balanced trade-off between local stability and global prediction accuracy. Additionally, the computed weights provided an interpretable measure of the relative contribution of each algorithm within the ensemble. 2.3. Model Performance Evaluation To comprehensively evaluate the prediction accuracy of the ensemble model, a set of metrics reflecting different aspects of error was applied (scikit-learn, 2025e). The mean absolute error (MAE) characterizes the average deviation of predicted values from experimental ones, regardless of their sign. The mean squared error (MSE) accounts for squared deviations and is sensitive to occasional large errors. The coefficient of determination ( R 2 ) measures the proportion of variance in the target variable explained by the model. The mean absolute percentage error (MAPE) represents the relative magnitude of errors in percentage terms, facilitating easier interpretation of the results. The use of multiple metrics provided a comprehensive assessment of accuracy and offered a more complete characterization of the model’s overall performance. 3. Results and discussion 3.1. Weight Contribution of Base Models Figure 1 presents the results of calculating the weights of the base models within the VotingRegressor ensemble algorithm, determined as the inverse of the mean squared error (1/MSE) for different cyclic loading frequencies. The analysis shows that the Gradient Boosting and MLP models consistently received the highest weights, indicating their strong effectiveness in capturing nonlinear relationships among stress, strain, and the number of loading cycles. The contributions of the Random Forest and Extra Trees models also remained significant, demonstrating their ability to generalize complex patterns even under increasing loading frequencies. In contrast, the SVR models consistently exhibited the lowest weights, reflecting their limited suitability for this task. The influence of the KNN model was moderate but tended to increase with higher frequencies. 16912 3051 18573