The Feynman-Kac Operator Expectation Estimator (FKEE) is an innovative method for estimating the target Mathematical Expectation $\mathbb{E}_{X\sim P}[f(X)]$ without relying on a large number of samples, in contrast to the commonly used Markov Chain Monte Carlo (MCMC) Expectation Estimator. FKEE comprises diffusion bridge models and approximation of the Feynman-Kac operator. The key idea is to use the solution to the Feynmann-Kac equation at the initial time $u(x_0,0)=\mathbb{E}[f(X_T)|X_0=x_0]$. We use Physically Informed Neural Networks (PINN) to approximate the Feynman-Kac operator, which enables the incorporation of diffusion bridge models into the expectation estimator and significantly improves the efficiency of using data while substantially reducing the variance. Diffusion Bridge Model is a more general MCMC method. In order to incorporate extensive MCMC algorithms, we propose a new diffusion bridge model based on the Minimum Wasserstein distance. This diffusion bridge model is universal and reduces the training time of the PINN. FKEE also reduces the adverse impact of the curse of dimensionality and weakens the assumptions on the distribution of $X$ and performance function $f$ in the general MCMC expectation estimator. The theoretical properties of this universal diffusion bridge model are also shown. Finally, we demonstrate the advantages and potential applications of this method through various concrete experiments, including the challenging task of approximating the partition function in the random graph model such as the Ising model.

翻译：费曼-卡克算子期望估计器（FKEE）是一种创新的方法，用于估计目标数学期望 $\mathbb{E}_{X\sim P}[f(X)]$，与常用的马尔可夫链蒙特卡洛（MCMC）期望估计器不同，它不依赖于大量样本。FKEE 包含扩散桥模型和费曼-卡克算子的近似。其核心思想是利用费曼-卡克方程在初始时刻的解 $u(x_0,0)=\mathbb{E}[f(X_T)|X_0=x_0]$。我们使用物理信息神经网络（PINN）来近似费曼-卡克算子，这使得扩散桥模型能够被整合到期望估计器中，显著提高了数据使用效率并大幅降低了方差。扩散桥模型是一种更通用的 MCMC 方法。为了整合广泛的 MCMC 算法，我们提出了一种基于最小 Wasserstein 距离的新扩散桥模型。该扩散桥模型具有普适性，并减少了 PINN 的训练时间。FKEE 还减轻了维度诅咒的不利影响，并弱化了通用 MCMC 期望估计器中关于 $X$ 的分布和性能函数 $f$ 的假设。本文也展示了这种通用扩散桥模型的理论性质。最后，我们通过各种具体实验证明了该方法的优势和潜在应用，包括在随机图模型（如 Ising 模型）中近似配分函数这一具有挑战性的任务。