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。

费曼 - 卡克运算符期望估计器

Feynman-Kac Operator Expectation Estimator

Physically informed neural networks (PINNs) are a promising emerging method
for solving differential equations. As in many other deep learning approaches,
the choice of PINN design and training protocol requires careful craftsmanship.
Here, we suggest a comprehensive theoretical framework that sheds light on this
important problem. Leveraging an equivalence between infinitely
over-parameterized neural networks and Gaussian process regression (GPR), we
derive an integro-differential equation that governs PINN prediction in the
large data-set limit -- the Neurally-Informed Equation (NIE). This equation
augments the original one by a kernel term reflecting architecture choices and
allows quantifying implicit bias induced by the network via a spectral
decomposition of the source term in the original differential equation.

物理信息神经网络是一种有效求解偏微分方程的新方法，通过理论框架将其与高斯过程回归等价，并推导出由其架构选择所引起的核项来增强其预测能力，并通过源项的谱分解量化其隐含偏差