PSI - Issue 79

Qinghui Huang et al. / Procedia Structural Integrity 79 (2026) 291–297

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The simple network shows rapid declines in both training and validation losses during the initial 200 epochs, but stabilizes beyond this threshold with validation losses slightly higher than training losses. This indicates limited capability to capture nonlinear relationships and evidence of underfitting. The single-hidden-layer network demonstrates continuous decreases in both training and validation losses through increased epochs, with validation losses approaching training losses. Notably, when epochs reach 800-1000, validation losses hit their lowest point before stabilizing, suggesting a strong learning capacity to effectively model nonlinear relationships between input parameters and incremental plastic strain, while early stopping techniques successfully mitigated overfitting. The double-hidden-layer network maintains an overall downward trend in training losses, though with fluctuations consistently exceeding those of the single-hidden-layer network. This discrepancy likely stems from its higher complexity and insufficient training sample size to support effective model learning. Fig. 6 and Fig. 7 present the prediction results and performance metrics of three structural neural network models (simple network, single-hidden layer, and double-hidden-layer) on the validation set. In the scatter plot comparing predicted values with actual ones, the single-hidden-layer model achieved a validation set R² value of 0.9797. This indicates great alignment between predictions and actual values, demonstrating the ability to accurately capture the plastic strain accumulation. In contrast, the simple network (R²=0.5449) showed more scattered data points around the diagonal, while the double-hidden layer model (R²=0.8963) exhibited lower consistency in data distribution compared to the single-hidden-layer model. This suggests that the simple network lacks the complexity required to model nonlinear relationships, whereas the double-hidden-layer model may have been affected by overfitting or improper architecture selection.

Figure 6 Predicted results under validation set.

Figure 7 RMSE, R ² and MAE values (under validation set) for different model structures

The bar charts of quantitative indicators such as root mean square error (RMSE), R², and mean absolute error (MAE) further quantify model performance. Regarding RMSE, the single-hidden-layer model (Model 2) achieved the lowest value of 0.0003, significantly lower than the simple network (Model 1: 0.0015) and the double-hidden-layer (Model 3: 0.0007). A lower RMSE indicates smaller deviations between predicted and actual values. For R², the single-hidden layer model reached 0.9797, markedly higher than the simple network (0.5449) and the double-hidden-layer (0.8963). A higher R² value signifies stronger variance interpretation capability and superior predictive performance. In terms of mean absolute error (MAE), the single-hidden-layer model demonstrated the best performance with a value of