We propose a method utilizing physics-informed neural networks (PINNs) to solve Poisson equations that serve as control variates in the computation of transport coefficients via fluctuation formulas, such as the Green--Kubo and generalized Einstein-like formulas. By leveraging approximate solutions to the Poisson equation constructed through neural networks, our approach significantly reduces the variance of the estimator at hand. We provide an extensive numerical analysis of the estimators and detail a methodology for training neural networks to solve these Poisson equations. The approximate solutions are then incorporated into Monte Carlo simulations as effective control variates, demonstrating the suitability of the method for moderately high-dimensional problems where fully deterministic solutions are computationally infeasible.

翻译：我们提出一种利用物理信息神经网络（PINNs）求解泊松方程的方法，该方程在通过涨落公式（如Green-Kubo公式和广义爱因斯坦型公式）计算输运系数时充当控制变量。通过采用神经网络构建的泊松方程近似解，我们的方法显著降低了当前估计量的方差。我们对估计量进行了全面的数值分析，并详细阐述了训练神经网络求解这些泊松方程的方法。随后将近似解作为有效控制变量纳入蒙特卡洛模拟，证明该方法适用于完全确定性求解计算不可行的中高维问题。