AI 中文总结
该研究针对带随机输入的动力学方程不确定性量化难题，提出用神经代理替代经典降阶模型的多保真度控制变量方法，证明其无偏性与方差稳定性，扩展至多控制变量及渐近保形层次结构，经数值测试验证有效性。
AI 中文摘要
对带有随机输入的动力学方程进行高效不确定性量化极具挑战性，因为这需要反复模拟高维模型，如玻尔兹曼方程、朗道方程及相关碰撞方程，其计算成本会迅速变得高昂。多保真度控制变量通过将少量高保真度模拟与对复杂度更低的降阶模型的大量评估相结合，解决了这一难题。在本研究中，我们分析了将降阶模型替换为神经代理而非通过经典数值格式进行评估的情况。我们证明，所得估计量仍为无偏估计，且神经近似引起的最优方差变化由精确低保真度可观测量与其神经近似之间的误差控制。该估计随后与非均匀福克-普朗克方程及Bhatnagar-Gross-Krook代理的残差稳定性估计相结合。我们还将分析扩展到多个控制变量以及包含极限欧拉可观测量的渐近保形（AP）层次结构。在流体极限下，最优层次方差收敛到与极限欧拉控制相关的方差，而中间动力学修正的贡献消失。基于微观-宏观神经代理的数值测试验证了所预测的方差稳定性及AP层次结构的行为。
英文摘要
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.
