# Machine Learning for the Prediction of Thermodynamic Properties in Amorphous Silicon
## 3 Results
### Predictive Models  
As discussed in the previous section, predictive models are essential for obtaining information about the system's state under different conditions. To achieve this, models were constructed using the \(R_{c}\), \(T\), and \(P\) as predictors and the \(\rho\), \(U\), and \(H\) as responses. The models were trained and tested using five-fold cross-validation, where the dataset was divided into five equal groups. Four groups were used as the training set, and the remaining group served as the testing set. This procedure was repeated, with each group  
Figure 3: (**a**–i) Relationships between the output variables (density (\(\rho\)), internal energy (\(U\)), enthalpy (\(H\))) and the input variables (cooling rate (\(R_{c}\)), temperature (\(T\)), pressure (\(P\))). Each dot represents a single MD simulation. While slight trends are observed for \(\rho\), \(U\), \(H\) with \(R_{c}\), pronounced trends become evident when analyzed as functions of \(T\) and \(P\). used as the testing set in turn. The model performance was evaluated by calculating the coefficient of determination (\(R^{2}\)) and root-mean-square error (RMSE) across all the test sets as follows: First, the three input variables (\(R_{c}\), \(T\), and \(P\)) calculated for a given a-Si sample were used by the regression model to predict the values of the density, internal energy, and enthalpy. Then, these predicted values were compared with the actual values obtained from the dataset to compute \(R^{2}\) and RMSE. The predicted and actual values are shown in Figure 4a-c for the \(\rho\), Figure 4d-f for the \(U\), and Figure 4g-i for the \(H\), with the standardized variables reverted to their original units for better physical interpretability. In all the cases, the models exhibited excellent performance, where the \(R^{2}\) values exceeded 0.95 and the RMSE values were close to zero, indicating a high predictive accuracy. No significant differences were observed between the three models, suggesting that all approaches were highly effective at predicting the thermodynamic states of the a-Si. A notable observation was that the values in the plots appeared grouped due to the discretized ranges used for the input variables. This grouping was particularly evident for the \(H\), as the pressure (\(P\)) was explicitly involved in the calculation of the enthalpy. Exploration of the residuals of the Linear, Ridge, and SVR models can be found in the Supplementary Materials. While the focus of this study was on the Linear, Ridge, and SVR models, other models were also explored and evaluated using the same cross-validation procedure. Specifically, Lasso regression and Elastic Net were considered. These models extend Ridge regression by incorporating penalty terms into the cost function, which can assist with feature selection and handling highly correlated predictors. However, both the Lasso and Elastic Net models showed lower performances and were excluded from further analysis (see the Supplementary Materials). To demonstrate the scope of the predictive models developed in this study, we performed predictions for \(\rho\), \(U\), and \(H\) based on the variables co  
Figure 4: (**a–i**) Predictive capability for the density (\(\rho\)), internal energy (\(U\)), and enthalpy (\(H\)) of the Linear, Ridge, and Support Vector Regression models using the testing set. The subscript \(p\) corresponds to the predicted values and \(a\) to the actual values. and pressure using the Linear regression model, which was chosen for its simplicity and interpretability. The following equation was employed:  
\[\rho=(0.015)\cdot R_{c}-(0.159)\cdot T+(0.984)\cdot P+0, \tag{7}\]  
where all the predictors were standardized to a mean of zero and a variance of one. This model was used to generate predictions that were incorporated into the plots shown in Figure 3 by reversing the input and output values to their original units of measure, providing a visual representation of the relationships between the \(\rho\) and the input variables.