## 3 Experimental Setup
### Ablation Study
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Model Variant**}} & \multicolumn{3}{c}{**ICME Dataset**} & \multicolumn{3}{c}{**Materials Project Dataset**} \\  & **PSNR** & **SSIM** & **MAE** & **AUC** & **PSNR** & **SSIM** & **MAE** & **AUC** \\ \hline w./o. Hybrid Stress-Strain Modeling Framework & \(33.76\pm 0.02\) & \(0.870\pm 0.02\) & \(0.067\pm 0.01\) & \(0.899\pm 0.02\) & \(33.04\pm 0.02\) & \(0.865\pm 0.01\) & \(0.064\pm 0.01\) & \(0.883\pm 0.02\) \\ w./o. Adaptive Learning with Physics-Consistency & \(34.58\pm 0.02\) & \(0.885\pm 0.02\) & \(0.061\pm 0.01\) & \(0.907\pm 0.01\) & \(34.12\pm 0.02\) & \(0.879\pm 0.02\) & \(0.060\pm 0.01\) & \(0.892\pm 0.01\) \\ w./o. Iterative Error Minimization & \(33.21\pm 0.02\) & \(0.872\pm 0.01\) & \(0.065\pm 0.01\) & \(0.894\pm 0.02\) & \(33.58\pm 0.02\) & \(0.869\pm 0.01\) & \(0.063\pm 0.01\) & \(0.878\pm 0.02\) \\ Full Model (Ours) & \(\mathbf{36.45\pm 0.02}\) & \(\mathbf{0.902\pm 0.02}\) & \(\mathbf{0.054\pm 0.01}\) & \(\mathbf{0.928\pm 0.02}\) & \(\mathbf{35.18\pm 0.02}\) & \(\mathbf{0.890\pm 0.02}\) & \(\mathbf{0.057\pm 0.01}\) & \(\mathbf{0.905\pm 0.02}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Ablation study results on model variants across ICME and Materials Project datasets. Figure 8: Ablation study of our method on the ICME and Materials Project datasets. Figure 7: Ablation study of our method on JARVIS-DFT and Zenodo datasets. indicates that the proposed approach exhibits high accuracy and stability in stress-strain prediction for structural mechanics. The results suggest that the model effectively reduces prediction errors under different data distributions, improving the accuracy of stress-strain modeling while demonstrating superior adaptability to variations in material properties and data patterns compared to conventional deep learning methods. Figure 9 presents a comparative error analysis across multiple models on different datasets, highlighting the accuracy and robustness of the proposed approach. The experimental results demonstrate that our proposed SSAPM model consistently outperforms existing state-of-the-art methods, including the recently introduced DeepH and MAT-Transformer models. In Table 6, across both the JARVIS-DFT and Zenodo datasets, SSAPM achieves the highest PSNR and SSIM values while maintaining the lowest RMSE, indicating its superior ability to reconstruct and predict stress-strain relationships with high fidelity. The AUC values further confirm the model's robustness in classification tasks, with SSAPM outperforming DeepH and MAT-Transformer by noticeable margins. Notably, DeepH, which employs a hybrid deep learning approach integrating physical constraints, performs competitively but still falls short in terms of predictive accuracy and noise resilience. MAT-Transformer, designed to leverage attention-based feature extraction for stress-strain modeling, exhibits strong generalization but does not surpass our method in overall performance. The improvements achieved by SSAPM can be attributed to its hybrid modeling framework, which effectively integrates physics-informed constraints with deep learning representations. Unlike DeepH, which primarily relies on a hybrid neural network for prediction, SSAPM dynamically adapts to material variations through its adaptive optimization strategy, allowing it to generalize better across different material datasets. Compared to MAT-Transformer, which depends heavily on attention-based mechanisms, SSAPM incorporates multi-sensor fusion techniques that enrich its feature representation, leading to more precise and stable predictions. The significant reduction in RMSE highlights SSAPM's ability to mitigate error propagation, ensuring more reliable stress-strain estimations. These results reinforce the effectiveness of combining mechanistic modeling with data-driven corrections, further positioning SSAPM as a leading approach in the field of structural mechanics and material modeling.