Feynman-Kac Operator Expectation Estimator
Jingyuan Li, Wei Liu
TL;DR
The paper introduces Feynman-Kac Operator Expectation Estimator (FKEE), a neural network-based method that efficiently estimates expectations with fewer samples, reducing variance and addressing high-dimensional challenges.
Contribution
It proposes a novel FKEE method using PINNs and diffusion bridge models, including a new Wasserstein-based diffusion bridge, improving efficiency and applicability over traditional MCMC methods.
Findings
FKEE significantly reduces variance compared to MCMC.
The method effectively approximates partition functions in complex models.
The universal diffusion bridge model accelerates training and broadens applicability.
Abstract
The Feynman-Kac Operator Expectation Estimator (FKEE) is an innovative method for estimating the target Mathematical Expectation 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 . 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…
Peer Reviews
Decision·Submitted to ICLR 2025
The goals stated in the introduction are bold, and quite interesting. Obtaining results in this line would prove quite useful in general for ML and statistics. I appreciate that the authors include small introductions for the Euler-Maruyama method and physics informed neural networks in the appendix. However their existence should be indicated in the main text.
Presentation is bad throughout. There are plenty of typos. The authors do not use parenthetical citations and instead insert them in the text which makes for a less pleasant reading experience. The notation introduced in line 232 definitely needs improvement, I do not understand which side is supposed to be the one that will be used later. Even then, it is unclear what is being defined, as there are two definitions for $\hat{\mu}\_{t\_i}$ . In Assumption 2.2, it is not clear what $\mu^{\mathc
A significant strength of this work is the innovative linking of the diffusion model to high-dimensional partial differential equations (PDEs), with Physics-Informed Neural Networks (PINNs) effectively employed to overcome the curse of dimensionality in solving these PDEs.
1. The Feynman–Kac model (Algorithm 2) with the PINN solver lacks a convergence or error estimate, which would be valuable for assessing its accuracy and reliability. 2. In the experiments, the authors claim that "using fewer points on the Markov chain achieves higher accuracy in approximating expectations." However, it is unclear if this result generalizes beyond the specific example provided, as it appears quite context-dependent.
the idea of using Feyman-Kac to approximate the expectation is interesting.
The scalability of the algorithm w.r.t. dimension is not verified sufficiently. d=20 is too small. There are no real-world simulations. The authors criticize the large variance issue by the MCMC method but fail to justify theoretically why the proposed method yields a lower variance. The empirical support is limited. NIT: Theorem 2.1: the discretization error by Growall inequality is weak and exponentially dependent on time. Girsanov can be used to fix it.
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Taxonomy
TopicsComputational Physics and Python Applications
MethodsDiffusion