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.

该研究提出了 Feynman-Kac 操作符期望估计器以及扩展了传统的 Diffusion Bridge 模型，结合 Physically Informed Neural Networks 技术，实现了用较少样本估计目标数学期望的目的，并减少了维度灾难和对分布假设的依赖。