Issue 78

C. Popa et alii, Fracture and Structural Integrity, 78 (2026) 118-128; DOI: 10.3221/IGF-ESIS.78.07

To evaluate model performance objectively, the dataset was partitioned into two independent subsets: 1) training set (80%), used to estimate model parameters; 2) testing set (20%), used to evaluate predictive capability on unseen data. The partitioning was performed using random sampling with a fixed initialization value (random_state = 42) to guarantee reproducibility. The split operation is represented as:     , , , , train test train test X y X X y y  (1) Performance evaluation was conducted separately on both subsets to assess model generalization and reduce the risk of overfitting. Linear Regression for Optimal Shape Prediction To predict optimal shape responses, a supervised machine learning model based on Multiple Linear Regression (MLR) [17] was implemented. The model assumes that the target output can be represented as a linear combination of the selected input variables. where: MLR S is the predicted output value; 0  is the intercept; 1 2 3 , ,    are learned coefficients; , , A B C are the input variables. The regression model was trained using the Ordinary Least Squares (OLS) optimization approach implemented through the Scikit-learn library. Prediction quality was evaluated using three statistical indicators [18]: Coefficient of Determination (R²) 0 1 2 3 MLR S A B C         (2)

2

ˆ y y

(   (   i

)

2

i ˉ

1  

R

(3)

2

y y

)

i

measures the proportion of variance explained by the model. Root Mean Square Error

1 2





 2

ˆ y y 



RMSE

(4)

i

i

measures prediction error magnitude. Mean Absolute Error

1 2

ˆ i MAE y y    i

(5)

measures average absolute deviation. After validation, the trained model was applied to newly generated combinations of input variables to estimate additional candidate shapes and support optimal shape selection. This workflow enables efficient exploration of the design space while reducing the need for additional simulations or experiments. To summarise we used 116 simulations for training and 29 for validation, thus the data split was 80%-20%. After training and validating the model, we used it to predict optimal shape for 85 new geometries. Linear regression prediction results The Linear regression prediction results are presented in Fig. 6. The first plot (Fig. 6a) compares Actual (X-axis) vs Predicted (Y-axis) values for the training set. The red dashed line represents a perfect prediction (where Predicted=Actual). The

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