Stochastic hierarchical data-driven optimization: application to plasma-surface kinetics

TL;DR

The paper tackles calibrating high-dimensional, sloppy physical models when simulations are expensive and data are scarce. It introduces a stochastic hierarchical optimization that identifies stiff parameter directions via a reduced Gauss-Newton Hessian and navigates the loss landscape with a heuristic meta-simulator, underpinned by a probabilistic loss derived from observed data. Applied to a plasma-surface kinetics model for atomic oxygen recombination on Pyrex, the method achieves rapid, sample-efficient convergence (validated on 225 conditions, with test R^2 ≈ 0.736) and yields uncertainty estimates that highlight the most influential parameters. This approach offers a transferable, scalable tool for inverse problems in complex reaction systems and provides a principled way to quantify parameter stiffness and predictive uncertainty in data-sparse regimes.

Abstract

This work introduces a stochastic hierarchical optimization framework inspired by Sloppy Model theory for the efficient calibration of physical models. Central to this method is the use of a reduced Hessian approximation, which identifies and targets the stiff parameter subspace using minimal simulation queries. This strategy enables efficient navigation of highly anisotropic landscapes, avoiding the computational burden of exhaustive sampling. To ensure rigorous inference, we integrate this approach with a probabilistic formulation that derives a principled objective loss function directly from observed data. We validate the framework by applying it to the problem of plasma-surface interactions, where accurate modelling is strictly limited by uncertainties in surface reactivity parameters and the computational cost of kinetic simulations. Comparative analysis demonstrates that our method consistently outperforms baseline optimization techniques in sample efficiency. This approach offers a general and scalable tool for optimizing models of complex reaction systems, ranging from plasma chemistry to biochemical networks.

Figures (4)

• Figure 1: Hierarchical algorithm overview diagram
• Figure 2: Optimization performance and geometric analysis. Figure (a) compares the convergence of the training loss ($\Phi_{\rm train}$) versus simulator iterations for the hierarchical methods (brown and pink) against the baseline algorithms, where the dotted line marks the default parameter loss. Figure (b) presents the sample efficiency in the early optimization phase (first 50 iterations) for the Hierarchical Stochastic strategy using varying subspace sizes $k$. Figure (c) displays the simultaneous evolution of the training loss (dotted lines) and test loss (solid lines) for the best-performing algorithms (Hierarchical and TRF). Figure (d) shows the eigenvalue spectrum ($\{ \lambda_i \}$) of the Gauss-Newton Hessian, $H_{\rm GN}$, as a function of the eigendirections $V_i$ at the minimum $\theta_{\rm final}$ found. The $H_{\rm GN}$ is represented as $H_{\rm GN} = V \Lambda V^T$, with $\Lambda= \mathrm{diag}(\lambda_1, \dots,\lambda_n)$. In this case, $V_0$ corresponds to the stiff subspace, while the remaining directions $V_i$ are the sloppy subspace.
• Figure 3: Validation of the optimized kinetics model. (a) Parity plot of predicted vs. experimental $\gamma_O$ for the test sets of all 5 independent splits, showing a strong correlation ($R^2=0.736$). (b) Detailed comparison of $\gamma_O$ as a function of wall temperature for selected test conditions at three different pressures, highlighting the accuracy gain of the optimized parameters (diamonds) over the default values (squares).
• Figure 4: Comparison between the normalized default literature parameters (green) and the median optimized values (orange) obtained from 5-fold cross-validation. The numeric labels correspond to the steric factor indices (e.g., the number 30 corresponds to $k_{0,30}$, following the sequential reaction order of the full kinetic scheme. The error bars correspond to the Hessian-derived uncertainty bounds. Colored bars highlight the stiff parameters (Gray: Desorption; Blue: CO chemisorption; Pink: Metastable energies), while uncolored bars indicate sloppy parameters.