# Feynman-Kac Operator Expectation Estimator

arXiv: https://arxiv.org/abs/2407.02010 · HTML: https://mlsanity.com/paper/2407.02010/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 58 references that are in the map.
- First posted on arXiv 2 Jul 2024.

## Builds on (most cited first, 30 of 32)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
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- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Design](https://mlsanity.com/paper/1902.00275/) (2019)
- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Let us Build Bridges: Understanding and Extending Diffusion Generative Models](https://mlsanity.com/paper/2208.14699/) (2022)
- [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017)
- [Improved Variational Autoencoders for Text Modeling using Dilated Convolutions](https://mlsanity.com/paper/1702.08139/) (2017)
- [Efficient and Accurate Gradients for Neural SDEs](https://mlsanity.com/paper/2105.13493/) (2021)
- [Discrete Flows: Invertible Generative Models of Discrete Data](https://mlsanity.com/paper/1905.10347/) (2019)
- [Control functionals for Monte Carlo integration](https://mlsanity.com/paper/1410.2392/) (2014)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [Banach Wasserstein GAN](https://mlsanity.com/paper/1806.06621/) (2018)
- [Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learning](https://mlsanity.com/paper/2106.09004/) (2021)
- [MCMC Variational Inference via Uncorrected Hamiltonian Annealing](https://mlsanity.com/paper/2107.04150/) (2021)
- [High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm](https://mlsanity.com/paper/1908.10859/) (2019)
- [Non-convex Learning via Replica Exchange Stochastic Gradient MCMC](https://mlsanity.com/paper/2008.05367/) (2020)
- [Langevin diffusions and the Metropolis-adjusted Langevin algorithm](https://mlsanity.com/paper/1309.2983/) (2013)
- [Accelerated Training of Physics-Informed Neural Networks (PINNs) using Meshless Discretizations](https://mlsanity.com/paper/2205.09332/) (2022)
- [Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learning](https://mlsanity.com/paper/2011.04602/) (2020)
- [Gradient-free Hamiltonian Monte Carlo with Efficient Kernel Exponential Families](https://mlsanity.com/paper/1506.02564/) (2015)
- [Unbiased Markov chain Monte Carlo with couplings](https://mlsanity.com/paper/1708.03625/) (2017)
- [Semi-Exact Control Functionals From Sard's Method](https://mlsanity.com/paper/2002.00033/) (2020)

## Abstract (arXiv)

The Feynman-Kac Operator Expectation Estimator (FKEE) is an innovative method for estimating the target Mathematical Expectation mathbbEX~ 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(x0,0)=mathbbE[f(XT)|X0=x0]. 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.