Machine learning models for Si nanoparticle growth in nonthermal plasma

TL;DR

The paper addresses the computational bottleneck of modeling nanoparticle growth in nonthermal plasmas by predicting the sticking probability $P_ ext{st}$ for Si-containing collisions using machine learning trained on classical reactive MD data. It systematically evaluates seven permutation-invariant ML approaches with tailored losses—favoring binomial NLL and its logit formulations—across nested cross-validation schemes to gauge generalization to unseen temperatures, clusters, and impactors. Key findings show that high predictive accuracy can be achieved with only 15–25% of the data, and that DeepSets and LGBM offer strong extrapolation capabilities for unseen temperatures and structures, respectively, with permutation invariance significantly boosting robustness. The approach substantially reduces MD cost for deriving growth parameters in nonthermal plasmas and is readily adaptable to other NP growth contexts, enabling more efficient and realistic reactor-scale simulations.

Abstract

Nanoparticles (NPs) formed in nonthermal plasmas (NTPs) can have unique properties and applications. However, modeling their growth in these environments presents significant challenges due to the non-equilibrium nature of NTPs, making them computationally expensive to describe. In this work, we address the challenges associated with accelerating the estimation of parameters needed for these models. Specifically, we explore how different machine learning models can be tailored to improve prediction outcomes. We apply these methods to reactive classical molecular dynamics data, which capture the processes associated with colliding silane fragments in NTPs. These reactions exemplify processes where qualitative trends are clear, but their quantification is challenging, hard to generalize, and requires time-consuming simulations. Our results demonstrate that good prediction performance can be achieved when appropriate loss functions are implemented and correct invariances are imposed. While the diversity of molecules used in the training set is critical for accurate prediction, our findings indicate that only a fraction (15-25\%) of the energy and temperature sampling is required to achieve high levels of accuracy. This suggests a substantial reduction in computational effort is possible for similar systems.

Figures (18)

• Figure 1: B-NLL, L-MSE, and logit-transformed Huber (L-H) losses rescaled and aligned for true $P_\mathrm{st}$s of a) 0.1, b) 0.5, c) 0.9.
• Figure 2: Temperature dependence of the sticking probability for collisions between different Si2H_y and a)Si4, b)Si2H6, and c)Si29H36. Lines show the fitted trend, described in (\ref{['eq:fit']}). Error bars represent two standard deviations. Si2H_y indicate molecules with balanced hydrogen distribution, while unbalanced fragments are expanded for clarity.
• Figure 3: Sticking probability vs. temperature for the collisions between Si2H6 and Si29 cluster with different hydrogen coverages of Si29H_x. The line represents the fit discussed in (\ref{['eq:fit']}), and error bars (generally smaller than the symbols) represent two standard deviations.
• Figure 4: Performance of 5-fold cross-validation. a) and b) show the average performance of each model trained and evaluated using the adjusted B-NLL and L-MSE, respectively. Black bars indicate the standard deviation across all five folds and five random seeds. c) and d) show the LGBM predictions for all folds and seeds using the same loss functions as a) and b), respectively. e) shows the adjusted B-NLL performance of the permutation invariant (dark blue) and variant (light blue) LGBM models and the naïve model as the fraction of training data decreases. The shaded region indicates the standard deviation across random seeds.
• Figure 5: Performance of leave-/one-/cluster-/out CV. a) and b) show the average performance of each model trained and evaluated using the adjusted B-NLL and L-MSE, respectively. Black bars indicate the standard deviation across random seeds. c) and d) show the predictions of LGBM for all folds and random seeds using the same loss functions as a) and b), respectively. Predictions for Si29H18 are highlighted in red. The losses for SVR Si29H31 and Si29H36 in b) are 23.5 and 41.5 but are truncated for visualization purposes.