Noise-balanced multilevel on-the-fly sparse grid surrogates for coupling Monte Carlo models into continuum models with application to heterogeneous catalysis

TL;DR

The paper tackles the computational bottleneck of coupling high-fidelity Monte Carlo models to continuum-scale simulations by introducing a noise-balanced, multilevel on-the-fly sparse grid surrogate (ML-OTF-SG). It balances discretization and sampling errors to allocate Monte Carlo effort per data point efficiently, while preserving self-consistency during on-the-fly surrogate construction with a single hyperparameter $L$ controlling accuracy. Demonstrations on heterogeneous catalysis problems show high accuracy at modest costs, even in challenging 1p-kMC cases, highlighting the method's practical potential for integrating microscopic kinetics into macroscopic reactor models. This approach enables scalable, robust multiscale simulations with MC noise, offering significant impact for catalytic process design and operando CFD applications.

Abstract

Multiscale simulations utilizing high-fidelity, microscopic Monte Carlo models to provide the nonlinear response for continuum models can easily become computationally intractable. Surrogate models for the high-fidelity Monte Carlo models can overcome this but come with some challenges. One such challenges arise by the sampling noise in the underlying Monte Carlo data, which leads to uncontrolled errors possibly corrupting the surrogate even though it would be highly accurate in the case of noise-free data. Another challenge arises by the 'curse of dimensionality' when the response depends on many macro-variables. These points are addressed by a novel noise-balanced sparse grids interpolation approach which, in a quasi-optimal fashion, controls the amount of Monte Carlo sampling for each data point. The approach is complemented by a multilevel on-the-fly construction during the multiscale simulation. Besides its efficiency, a particularly appealing feature is the ease of use of the approach with only a single hyperparameter controlling the whole surrogate construction - from the surrogate's accuracy with guaranteed convergence to which data needs to be created with which accuracy. The approach is demonstrated on challenging examples from heterogeneous catalysis, coupling microscopic kinetic Monte Carlo models into macroscopic reactor simulations.

Figures (7)

• Figure 1: (a) One-dimensional basis functions $\phi_{l,i}$ of different levels $l$ used in the sparse grid construction. (b) Regular sparse grid in 2D. (c) On-the-fly constructed sparse grid, "on-the-fly" evaluation points are depicted as in blue.
• Figure 2: Illustration of multiscale modeling of heterogeneous catalysis in a fixed-bed reactor.
• Figure 3: a) Solutions for CO2 concentration of the coupled system \ref{['eq:fixed-bed-reactor-model_concentration']} with the analytical reverse-water-gas-shift model Equation \ref{['eq:rwgs-R']} and different level $L$ of on-the-fly interpolation for a noise STD $\sigma = 0.3$. Reference solution from direct coupling without interpolation is shown in the black dotted line. b) $L_2$ error of the noise balanced ML-OTF-SG scheme over grid size of the sparse grid for noise levels $\sigma=0.3$ and $\sigma=1$. c) $L_2$ error of the noise balanced ML-OTF-SG scheme over sum of samples $\sum_j N_j$ for noise levels $\sigma=0.3$ and $\sigma=1$ as well as theoretical limiting $O(N^{-1/2})$ scaling for Monte Carlo methods.
• Figure 4: a) Solutions for the CO partial pressure of the coupled system \ref{['eq:fixed-bed-reactor-model_concentration']} for the RuO2 CO-oxidation model (Section \ref{['sec:CO-oxidation-RuO2']}) and different levels $L$ of on-the-fly interpolation. b) Approximate computational cost (blues circles left axis) and number of grid points (black squares right axis)
• Figure 5: Turnover frequency of the RuO2 model over $p_{\ch{CO}}$, $p_{\ch{O2}}$ with $p_{\ch{O2}},p_{\ch{CO}} \in [1\times 10^{-5},1]$bar and $T=650K$ averaged over 10 samples per evaluation point. ML-OTF-SG solver query points are depicted in red, the corresponding grid points as black dots.