Feynman-Kac Operator Expectation Estimator

TL;DR

The paper presents FKEE, a framework that unifies diffusion-bridge modeling with Feynman-Kac PDE decoding to estimate $\mathbb{E}_{X\sim P}[f(X)]$ using far fewer samples than traditional MCMC. By learning a neural SDE that matches a target distribution and solving a Feynman-Kac PDE with a PINN, FKEE achieves variance reduction and better scalability in high dimensions, while remaining meshless and applicable to a broad class of $P$ and $f$. The approach is supported by theoretical analyses (consistency and error bounds) and demonstrates practical gains in Ising-model partition function estimation and other high-dimensional settings. The work promises substantial impact for efficient, PDE-informed probabilistic inference in statistics and machine learning, enabling robust expectation estimation beyond LLN/ETMC assumptions.

Abstract

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.

Figures (4)

• Figure 1: Comparison of the probability density functions of the generated and resampled paths and target distributions for each dimension. Two of the blue lines are the mean of the experience of the target sample and the mean of the experience of the re-generation sample, respectively.
• Figure 2: Generated tracks
• Figure 3: The empirical distribution of $\mathbb E_{estimated}(X_{10})$
• Figure 4: The empirical distribution of $\mathbb E_{estimated}(X_{1})$

Theorems & Definitions (20)

• Theorem 2.1
• proof
• Theorem 2.6
• proof
• Theorem 2.7
• proof
• Theorem 2.8
• proof
• Theorem 8.1
• proof