# Machine Learning for the Prediction of Thermodynamic Properties in Amorphous Silicon
## 3 Results
### Statistical Exploration  
The values obtained for the \(\rho\), \(U\), and \(H\) are summarized in the boxplots shown in Figure 2. The density ranged from 2.3 to 2.7 g/cm\({}^{3}\), with a median value close to 2.4 g/cm\({}^{3}\), which aligned well with previous findings on amorphous silicon [37; 38]. The observed lower and upper limits were primarily dictated by the temperature and pressure variations, as is further analyzed in subsequent sections. The internal energy spanned from \(-4.525\) to \(-4.375\) eV/atom, with a median near \(-4.44\) eV/atom. Since \(U\) is composed of both potential and kinetic energy components, it is indirectly influenced by pressure and temperature. Structural modifications induced by pressure affect the potential energy landscape, while temperature variations modulate atomic vibrations, increasing the kinetic energy [39; 40; 41]. Enthalpy exhibits a systematic shift toward higher values relative to internal energy due to the additional contribution of the pressure-volume term in its calculation, as defined by \(H=U+PV\). This term accounts for the work associated with maintaining a constant-pressure environment. The boxplots provided an effective means of statistically exploring the distribution of the thermodynamic variables. By visually representing the central tendency, spread, and presence of potential outliers, these boxplots allowed for an intuitive assessment of variability and underlying trends in the data. They also facilitated direct comparisons between different variables, which helped to identify how external factors, such as the temperature and pressure, influenced the system. Furthermore, the boxplots revealed a relatively uniform distribution of values across all the output variables, which can be attributed to the discrete nature of the input variables--\(R_{c}\), \(T\), and \(P\)--rather than stochastic sampling, as outlined in the methodology. While the boxplots provide an overview of the calculated values, they do not explicitly reveal the influence of the input variables \(R_{c}\), \(T\), and \(P\). To further explore these dependencies, scatter plots illustrating the relationships between the input and output variables were generated, as shown in Figure 3. Due to the discretized selection of input values, the data points appeared evenly distributed rather than randomly scattered. The density \(\rho\) exhibited a slightly decreasing trend with \(R_{c}\) and a more pronounced decline with increasing temperature \(T\). This behavior is consistent with previous studies indicating that higher cooling rates lead to an increased free volume, while elevated temperatures enhance atomic vibrations, effectively increasing the atomic spacing [42; 43]. Both effects contribute to a reduction in the structural compactness, resulting in lower densities. In contrast, \(\rho\) shows a clear increasing trend with pressure \(P\), as compression induces a more compact atomic arrangement [39]. For both the internal energy \(U\) and enthalpy \(H\), increasing trends were observed across all the input variables. Higher cooling rates led to higher energy states, which are often associated with structural rejuvenation in amorphous materials [44]. The effects of \(T\) and \(P\) on the \(U\) and \(H\) were more direct: increasing temperature raised the kinetic energy component of \(U\), while pressure explicitly influenced the \(H\) through the pressure-volume term in its definition. As a reminder, the internal energy is given by \(U=Pe+K\), where \(Pe\) represents the potential energy and \(K\) the kinetic energy. Meanwhile, enthalpy is defined as \(H=U+P\cdot V\), highlighting the contribution of work under constant pressure conditions. Figure 2: Boxplots for the output variables: (**a**) density (\(\rho\)), (**b**) internal energy (\(U\)), and (**c**) enthalpy (\(H\)). The bottom and top bars represent the minimum and maximum values, respectively. The bottom and top edges of the blue box correspond to the first and third quartile, respectively, while the orange bar represent the median.