# Machine Learning for the Prediction of Thermodynamic Properties in Amorphous Silicon
## 2 Materials and Methods
### MD Simulations  
The MD simulations were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS (version 29 August 2024) [24]). The atomic interactions  
Figure 1: Pipeline consisting of data collection from MD simulations, dataset preparation, and machine learning modeling for thermodynamic properties prediction. in the silicon were modeled with the Tersoff potential using the Si parameterization [25], which has been employed to study a-Si and related materials and effectively capture the covalent bonding characteristics of silicon; it has been extensively validated in previous studies on both crystalline and amorphous silicon [26; 27; 28; 29]. A time step of 1 fs was used throughout all the simulations to achieve a balance between numerical stability and computational efficiency to allow for the precise tracking of atomic motion while maintaining reasonable simulation times. A simulation box with dimensions of \(8.7\times 8.7\times 8.7\) nm\({}^{3}\) was constructed, which initially contained crystalline Si with a diamond-like structure. While these dimensions might not be large enough to study the mechanical deformation of a-Si or polycrystalline structures [12], they are suitable for studying thermodynamic properties [6], which was the focus of the current work. The chosen system size balanced computational efficiency with accuracy to ensure that the thermodynamic trends observed were representative of the bulk behavior while maintaining feasible simulation times. To prepare the amorphous state, the system was first minimized using the conjugate gradient algorithm to eliminate any unphysical atomic overlaps and reduce residual stresses in the initial crystalline structure. The temperature was ramped up to 3500 K at a rate of \(10^{13}\) K/s, followed by equilibration at 3500 K for 0.5 ns. Quenching was then performed at a cooling rate of \(R_{c}\) until reaching a temperature of 100 K. These steps were conducted under the NVT ensemble with periodic boundary conditions to eliminate surface effects and simulate bulk-like conditions. After quenching, the system was further equilibrated at a target temperature \(T\) and pressure \(P\) for 0.2 ns under the NPT ensemble. This relaxation step ensured that any residual stresses from rapid cooling were dissipated and that the system reached a stable thermodynamic state before the property calculations. To systematically explore the influence of the cooling rate, temperature, and pressure on the thermodynamic properties of the a-Si, a broad set of input conditions was considered. The input variables comprised five values for \(R_{c}\) (\(1\times 10^{11}\), \(5\times 10^{11}\), \(1\times 10^{12}\), \(5\times 10^{12}\), \(1\times 10^{13}\) K/s), five values for \(T\) (100, 200, 300, 400, and 500 K), and six values for \(P\) (0, 2, 4, 6, 8, and 10 GPa), which resulted in a total of 150 different simulation combinations that encompassed conditions that ranged from standard synthesis environments to high-pressure regimes where structural compaction is expected. These cooling rates were adopted based on previous studies that have shown that they effectively induce amorphization in covalent materials [30; 31], while the selected hydrostatic pressures are not high enough to cause plastic deformation of the sample [32]. For each final configuration, key thermodynamic quantities were extracted, namely, the density (\(\rho\)), internal energy (\(U\)), and enthalpy (\(H\)), which were selected as the output variables. The dataset generated from these 150 simulations provided a robust foundation for statistical analysis and the development of predictive models using traditional machine learning techniques, such as regression and decision trees [17]. While these methods are effective at identifying trends and interpolating within the sampled parameter space, more complex approaches, such as artificial neural networks, require significantly larger datasets for optimal training.