## 2 Methodology
### Multi-Objective AutoML Neural Network  
The AutoML framework AutoKeras was employed to generate neural network architectures automatically. AutoKeras, an open-source system built on the Keras deep learning API, streamlines model development by searching for optimal network architectures and hyperparameter configurations without manual intervention (Figure 2) [13]. Its architecture search and hyperparameter optimization capabilities are particularly advantageous for complex nonlinear regression tasks [14].  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Description** & **Value Range** \\ \hline \(E\) (GPa) & Initial contact stiffness & 1.5, 1.7, 1.9, 2.1, 2.2, 2.3, 2.5, 2.6, 2.7, 2.9, 3.2, 3.3, 3.4, \\ \(k_{s}/k_{n}\) & Shear-to-normal stiffness ratio & 3.5, 3.6, 3.7, and 4.0 \\ \(E_{d}\) (GPa) & Residual stiffness after bond breakage & 1.5, 1.7, 1.9, 2.1, 2.2, 2.3, 2.5, 2.6, 2.7, 2.9, 3.2, 3.3, 3.4, \\ \(k_{s,d}/k_{n,d}\) & Shear-to-normal stiffness ratio after bond failure & 1.0, 1.2, 1.4, 1.5, 1.7, 1.8, 2.0, 2.2, and 2.5 \\ \(\sigma_{t}\) (MPa) & Bond tensile strength & 5, 7, 9, 9.5, 10.0, 12.0, 15.0, 17.0, 17.5, 18.0, 20.0, 22.0, \\ \(c\) (MPa) & Bond cohesion (maximum shear stress) & 25.0, 26.0, 26.5, 27.0, 29.0, and 30.0 \\ \(\mu\) & Residual friction at broken bond contacts & 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Linear parallel bond model parameters employed in PFC3D simulations.  
Figure 2: Workflow of the AutoML algorithm.  
Unlike traditional neural network workflows, where hyperparameter tuning demands extensive trial and error, AutoKeras automates the selection of the model depth, layer types, activation functions, and regularization strategies. This automation ensures identification of the best-performing model with minimal human effort [13].  
A multi-objective regression model was constructed via AutoML to invert the PFC3D linear parallel bond parameters \(E\), \(k_{s}\)/\(k_{n}\), and \(E_{d}\) from the experimental stress-strain curves. The final layer consisted of a dense layer with three neurons, each predicting one of the target parameters. A custom-weighted mean squared error (MSE) loss function was defined to balance these outputs:  
\[MSE\ =\ \sum\omega_{i}(y_{i}\ -\ y_{i}^{*})^{2} \tag{1}\]  
Here, \(y_{i}\) and \(y_{i}^{*}\) denote the measured and predicted values for each parameter, respectively, and \(\omega_{i}\) represents the weight assigned to each output term. Weights were chosen according to the relative difficulty of predicting each parameter; optimal performance was achieved with \(\omega\) = [0.01, 0.10, 0.89] for \(E\), \(k_{s}\)/\(k_{n}\), and \(E_{d}\), respectively.  
The prediction accuracy of the multi-objective AutoML model was evaluated using the coefficient of determination, \(R^{2}\), defined as Equation (2). A value of \(R^{2}\) close to 1 indicates a better model performance.  
\[R^{2}=1-\frac{\sum_{i=1}^{n}(y_{i}-y_{i}^{*})^{2}}{\sum_{i=1}^{n}(y_{i}-\overline {y}_{i})^{2}} \tag{2}\]  
where \(y_{i}\) is the mean of the observed values. Values of \(R^{2}\) approaching unity indicate superior model performance.