Control variates with neural surrogates for uncertainty quantification in kinetic equations
TL;DR
This work analyzes the case in which the reduced model is replaced by a neural surrogate rather than evaluated through a classical numerical scheme and shows that the resulting estimator remains unbiased and that the change in the optimal variance induced by the neural approximation is controlled by the error between the exact low-fidelity observable and its neural approximation.
Abstract
Efficient uncertainty quantification for kinetic equations with random inputs is challenging because it requires repeated simulations of high-dimensional models, such as the Boltzmann, Landau, and related collisional equations, whose computational cost can quickly become prohibitive. Multifidelity control variates address this difficulty by coupling a small number of high-fidelity simulations with many evaluations of lower-complexity reduced models. In this work, we analyze the case in which the reduced model is replaced by a neural surrogate rather than evaluated through a classical numerical scheme. We show that the resulting estimator remains unbiased and that the change in the optimal variance induced by the neural approximation is controlled by the error between the exact low-fidelity observable and its neural approximation. This estimate is then combined with residual stability estimates for inhomogeneous Fokker--Planck and Bhatnagar--Gross--Krook surrogates. We also extend the analysis to several control variates and to an asymptotic-preserving (AP) hierarchy containing the limiting Euler observable. In the fluid limit, the optimal hierarchical variance converges to the variance associated with the limiting Euler control, while the contribution of the intermediate kinetic correction vanishes. Numerical tests based on micro--macro neural surrogates illustrate the predicted variance stability and the behavior of the AP hierarchy.