# Machine Learning for the Prediction of Thermodynamic Properties in Amorphous Silicon
## 4 Discussions  
The predictive models constructed in this study, which were based on the dataset generated by the MD simulations, demonstrated exceptional performances, as evidenced by \(R^{2}\) values that exceeded 0.95 and minimal root-mean-square errors. These results validate the accuracy and reliability of the models, confirming that they captured the complex relationships between the input variables (\(R_{c}\), \(T\), \(P\)) and the thermodynamic outputs, namely, density (\(\rho\)), internal energy (\(U\)), and enthalpy (\(H\)). More specifically, the models revealed that while the cooling rate (\(R_{c}\)) played a relatively minor role in determining these properties, it still influenced the energy state of the amorphous phase. In contrast, the temperature (\(T\)) and pressure (\(P\)) had significantly more pronounced impacts: the temperature affected the atomic mobility and energy distribution, while the pressure altered the atomic packing and density, which ultimately determined the material's behavior. This finding underscores the critical influences of thermal and pressure conditions on the thermodynamic state of a-Si, which is essential for understanding its behavior in various applications, including materials processing and device fabrication. In addition to validating the accuracy of the models, this study demonstrated the power of machine learning in generating additional thermodynamic data. The models enabled the construction of isobaric and isothermal curves, which provided valuable insights into the material's behavior under different thermodynamic conditions. These generated curves allowed for a deeper exploration of the thermodynamic properties of the a-Si, particularly by providing a means of examining the relationships between the material's  
Figure 5: **(a–i)** Predictions for the \(\rho\), \(U\), and \(H\) based on the \(R_{c}\), \(T\), and \(P\) using the Linear model. The dots correspond to data retrieved from the MD simulations. The dotted lines correspond to the predictions of the Linear model, where the thermodynamic parameter in the legend was kept fixed while the others varied in a given range. The black arrow in panel (**d**) denotes isotherms and isobars with comparable internal energies. These isotherms and isobars provide further insights into thermodynamic states. density, internal energy, and enthalpy at different fixed values of temperature or pressure. This ability to generate data points efficiently and systematically complemented traditional MD simulations, which can be computationally expensive and time-consuming, especially when exploring a broad range of thermodynamic states. By offering a more efficient pathway to explore these properties, the models contributed to a better understanding of the equation of state of the a-Si. Despite the impressive performance of the models within the training range, it is important to acknowledge their inherent limitations. The considered models were linear, and they were trained within a specific range of variables, and thus, their predictive accuracy outside this range has not been validated. Therefore, careful evaluation and validation of the models are necessary when applying them to thermodynamic conditions beyond the training data. This step is crucial to ensure the robustness and generalizability of the models. For instance, the models could be tested under more extreme or unconventional thermodynamic conditions, such as higher or lower temperatures and pressures, to assess their ability to make reliable predictions in these scenarios. Additionally, including data from additional phases, such as the liquid state, could improve the models and provide a more comprehensive description of phase transitions in a-Si. Predicting phase transitions, such as the transformation from amorphous to crystalline states or the glass-forming ability of the material, could be achieved by incorporating data from both the liquid and amorphous phases.