Feynman-Kac Operator Expectation Estimator: An Innovative Method for Enhancing MCMC Efficiency and Reducing Variance

Jingyuan Li, WEI LIU

Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

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Keywords: Expectation estimator, Diffusion bridge model, MCMC, PINN, Feynman-Kac formula

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TL;DR: We provide a new method to estimate the Mathematical Expectation by combining diffusion bridge models, PINN and Feynman-Kac operator, which improves the efficiency and reduces the variance.

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) algorithm. This method uses Physically Informed Neural Networks (PINN) to approximate the Feynman-Kac operator. It enables the incorporation of existing diffusion bridge models into the expectation estimator, and significantly improves the efficiency of using Markov chains while substantially reduces the variance. Additionally, this method mitigates the adverse impact of the curse of dimensionality, weakening the assumptions on the distribution of $X$ and $f$ in the general MCMC expectation estimator. In the algorithm implementation, the first step involves constructing a diffusion bridge over the target distribution or known data by matching the coefficients of the diffusion bridge from the random flow trajectories or a Markov chain. Subsequently, we employ PINN to solve the Feynman-Kac equation, and the solution of this equation provides the mathematical expectation in analytical form. 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.

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Submission Number: 4660