# Machine Learning for the Prediction of Thermodynamic Properties in Amorphous Silicon
## 2 Materials and Methods
### Predictive Models  
To develop the predictive models for estimating the thermodynamic properties of the a-Si, we employed Linear, Ridge, and Support Vector Regression (SVR) models [34; 35]. These models aimed to establish the mathematical relationships between the input variables (\(R_{c}\), \(T\), and \(P\)) and the output properties (\(\rho\), \(U\), and \(H\)) by facilitating the interpolation and potential extrapolation beyond the simulated data points. Linear regression operates by minimizing the sum of squared residuals to find the best-fit hyperplane that describes the relationships between the predictors and the response variables. While it is one of the most interpretable and widely used regression techniques, it can suffer from instability in the presence of multicollinearity, where predictor variables exhibit strong correlations. This issue may lead to large coefficient magnitudes, making the model sensitive to small variations in the input data and reducing its generalizability. To mitigate this issue, Ridge regression was also employed. Ridge regression introduces an L2 regularization term in the cost function, penalizing large coefficient values and shrinking them toward zero. This approach not only reduces overfitting but also improves the model robustness and lowers the variance, particularly when dealing with high-dimensional or correlated data. The Ridge regression model balances bias and variance by tuning the regularization parameter, ensuring better generalization to unseen data while preserving the predictive power of the model. SVR is based on the principles of Support Vector Machines. However, SVR estimates continuous outputs by minimizing a regularized loss function using an \(\epsilon\)-insensitive margin and support vectors. The three regression tasks were implemented using the Scikit-learn (version 1.4.2) library in Python (version 3.12.4) [36], which provides efficient tools for training, validating, and optimizing predictive models. The predictors \(R_{c}\), \(T\), and \(P\) were used to estimate the responses \(\rho\), \(U\), and \(H\) through the following expressions:  
\[\rho =\omega_{1,1}R_{c}+\omega_{1,2}T+\omega_{1,3}P+\beta_{1}, \tag{2}\] \[U =\omega_{2,1}R_{c}+\omega_{2,2}T+\omega_{2,3}P+\beta_{2},\] (3) \[H =\omega_{3,1}R_{c}+\omega_{3,2}T+\omega_{3,3}P+\beta_{3}, \tag{4}\]  
where \(\omega_{i,j}\) and \(\beta_{i}\) are the regression coefficients. A 5-fold cross-validation approach was employed to train and test the models. The hyperparameters used in this study can be found in the Supplementary Materials. The performance was assessed using the coefficient of determination \(R^{2}\) and the root-mean-square error (RMSE). \(R^{2}\) was computed using  
\[R^{2}=1-\frac{\sum_{i=1}^{n}(y_{a}-y_{p})^{2}}{\sum_{i=1}^{n}(y_{a}-\overline{y} )^{2}}, \tag{5}\]  
where \(n\) is the total number of observations (150 in this work), \(y_{a}\) is the actual value, \(y_{p}\) is the predicted value, and \(\overline{y}\) is the mean of \(y_{a}\). The \(R^{2}\) score ranges from zero (no predictive capability) to one (perfect prediction). The RMSE was calculated using  
\[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{\overline{n}}(y_{a}-y_{p})^{2}}, \tag{6}\]  
where higher RMSE values indicate a greater deviation between the actual and predicted values, while lower RMSE values reflect more accurate predictions. Although the RMSE is widely used due to its intuitive interpretation in the same units as the target variable, other performance metrics can also offer valuable insights. These include the mean absolute error (MAE), which measures the average absolute difference between the observed and predicted values; mean square error (MSE), which penalizes larger errors more heavily; and mean absolute percentage error (MAPE), which expresses errors as a percentage of actual values, making it useful for relative comparisons.