# Machine Learning for the Prediction of Thermodynamic Properties in Amorphous Silicon
## 3 Results
### Predictive Models
The dotted lines in Figure 5 represent the model's predictions, which allowed for a clear comparison between the predicted values and the original data from the MD simulations. In particular, the \(\rho\)-\(R_{c}\) relationship was examined by fixing \(P\) at 0 GPa (yielding the \(P=0\) GPa curves) and by fixing \(T\) at 100 K (resulting in the \(T=100\) K curves). This approach allowed for the construction of isobars and isotherms, respectively, which revealed deeper thermodynamic insights into the system's behavior. For example, varying the cooling rate in the \(\rho\) versus \(T\) and \(\rho\) versus \(P\) plots (represented by the red and green curves, respectively) resulted in closely spaced isobars and isotherms. This indicates that the \(R_{c}\) had a relatively minor influence on the thermodynamic states compared with the temperature and pressure, which had more pronounced effects. This is also distinguished in the (a), (d), and (g) panels, where the \(R_{c}\) is the independent variable. As observed, only the \(U\) varied significantly when the \(R_{c}\) increased, which is in agreement with previous works that have demonstrated that the cooling rate is explicitly related to the energy states of the sample [44; 45]. Additionally, while the \(H\) exhibited similar behavior to that of the \(\rho\), the \(U\) showed an interesting feature: in the \(U\) vs. \(R_{c}\) plot shown in panel (d), the isotherms overlap with the isobars. This suggests that increasing the pressure at a constant temperature significantly raised the internal energy, where it eventually equaled that of a state with slightly lower pressure but higher temperature. For instance, the curves for 100 K at 6 GPa and 200 K at 0 GPa (denoted by the black arrow) show comparable internal energy values. The same behavior is observed in the \(U\) vs. \(T\) plot. While both the \(U\) and \(H\) share the same units, the \(H\) did not display this behavior, which can be attributed to the pressure term in the definition of enthalpy, which resulted in more widely separated values for the \(H\). While the models accurately replicated the trends observed in the MD data for most variables, they did struggle to fully capture the slight decreasing trend in the \(\rho\)-\(R_{c}\) relationship. This discrepancy may be attributed to the fact that the relationship between the cooling rate and the density was more subtle and less consistent compared with the stronger trends observed for the temperature and pressure. In contrast, the models were particularly successful in capturing the decreasing trend in the \(\rho\)-\(T\) relationship and the increasing trend in the \(\rho\)-\(P\) relationship, as these patterns were more clearly defined in the data. These observations highlight the capabilities of the models to predict the thermodynamic states with a high degree of accuracy. However, it is important to note that the models were trained within a limited range of variables, and their ability to make accurate predictions outside of this range has not been validated. While the models performed well for the data within the specified ranges, further validation would be necessary to ensure that the predictions remain reliable and robust when applied to conditions outside the training set. Despite this limitation, the use of ML models presents a significant advantage by allowing for the generation of additional data points that can aid in the reproduction of the equation of state for a material system. Acquiring information for a single state through traditional MD simulations can be computationally expensive and time-consuming, especially when exploring a wide range of conditions. Therefore, ML provides a powerful and efficient tool for predicting thermodynamic properties over a broader spectrum of conditions, enabling faster exploration of the material's behavior. Additionally, by extending the approach to include data from both the liquid and amorphous phases, it may be possible to investigate the glass-forming ability of the material.