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

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.

💡 Research Summary

The paper addresses a fundamental bottleneck in multiscale simulations where a high‑fidelity microscopic Monte Carlo (MC) model is repeatedly queried by a macroscopic continuum solver. While surrogate models can alleviate the computational load, two intertwined difficulties arise: (i) the “curse of dimensionality” when the surrogate must approximate a response that depends on many macro‑variables, and (ii) the stochastic sampling noise inherent to MC evaluations, which can dominate the surrogate error if not properly controlled.

To tackle these issues, the authors propose a noise‑balanced multilevel on‑the‑fly sparse grid (ML‑OTF‑SG) framework. The method builds on hierarchical piecewise‑linear sparse‑grid interpolation, where a single hyper‑parameter – the maximal sparse‑grid level L – governs both the discretisation accuracy and the amount of training data to be generated. The key innovations are:

1. Multilevel on‑the‑fly construction – Instead of pre‑computing a full sparse grid, the surrogate is refined locally as the continuum solver queries new points. Early iterations use a coarse grid; as the solution approaches convergence, only the region of interest is enriched, dramatically reducing the number of required grid points.

2. Error decomposition and noise balancing – The total mean‑square error of the surrogate is split into a deterministic discretisation component (E_disc) that decays with L, and a stochastic component (E_noise) proportional to the variance of the MC estimator divided by the number of MC samples Nₗ used at each grid point. By enforcing the balance condition E_disc(L) ≈ E_noise(L, Nₗ), the algorithm automatically determines how many MC samples each point needs. In practice, higher‑level points receive many samples (to suppress noise where the grid is fine), while low‑level points are evaluated with few samples (since their contribution to the overall error is already limited).

3. Single‑parameter control – The user only specifies the maximal level L (or equivalently a target tolerance ε). The algorithm then self‑consistently selects the hierarchical refinement, the per‑point sample counts, and even a convergence test for the surrogate. This makes the approach “black‑box” friendly and removes the need for extensive hyper‑parameter tuning typical of machine‑learning surrogates.

4. Self‑consistent on‑the‑fly evaluation – Because the surrogate is constructed incrementally, any query to the same input at different solver iterations yields identical outputs, preserving the deterministic nature required for Newton‑type solvers and ensuring numerical stability.

The methodology is demonstrated on realistic heterogeneous catalysis problems. The authors couple first‑principles kinetic Monte Carlo (1p‑kMC) models, which simulate surface reaction events stochastically, with a stationary fixed‑bed reactor model. The input space (e.g., surface coverages, gas‑phase concentrations) can be 3‑5 dimensional, and each kMC evaluation is expensive and noisy. Using the noise‑balanced ML‑OTF‑SG surrogate, the authors achieve:

• Accuracy – Reaction rates and conversion predictions deviate by less than 1 % from the fully coupled MC‑direct solution.
• Efficiency – The total number of kMC evaluations is reduced by 70–80 % compared with a uniform sparse‑grid surrogate, and the overall wall‑clock time of the multiscale simulation drops to roughly 10–20 % of the naïve approach.
• Solver performance – The reduced noise leads to smoother Jacobian approximations, cutting the number of Newton iterations by about 30 %.

The paper concludes that the noise‑balanced ML‑OTF‑SG framework provides a principled, easy‑to‑use surrogate construction that simultaneously mitigates dimensionality and stochastic noise. The authors suggest extensions such as higher‑order basis functions, adaptive estimation of the balancing constant, and application to other stochastic HFMs (e.g., molecular dynamics). Overall, the work offers a compelling solution for integrating noisy, high‑cost microscale models into large‑scale continuum simulations without sacrificing robustness or accuracy.