# Deterministic Fokker-Planck Transport -- With Applications to Sampling, Variational Inference, Kernel Mean Embeddings & Sequential Monte Carlo

arXiv: https://arxiv.org/abs/2410.18993 · HTML: https://mlsanity.com/paper/2410.18993/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 76 references that are in the map.
- First posted on arXiv 11 Oct 2024.

## Builds on (most cited first, 17 of 17)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Probability flow solution of the Fokker-Planck equation](https://mlsanity.com/paper/2206.04642/) (2022)
- [A Unified Particle-Optimization Framework for Scalable Bayesian Sampling](https://mlsanity.com/paper/1805.11659/) (2018)
- [Stein Points](https://mlsanity.com/paper/1803.10161/) (2018)
- [Accelerated Information Gradient flow](https://mlsanity.com/paper/1909.02102/) (2019)
- [A Rigorous Theory of Conditional Mean Embeddings](https://mlsanity.com/paper/1912.00671/) (2019)
- [Stein Point Markov Chain Monte Carlo](https://mlsanity.com/paper/1905.03673/) (2019)
- [Sequential Kernel Herding: Frank-Wolfe Optimization for Particle Filtering](https://mlsanity.com/paper/1501.02056/) (2015)
- [Layered Adaptive Importance Sampling](https://mlsanity.com/paper/1505.04732/) (2015)
- [Kernel Conditional Density Operators](https://mlsanity.com/paper/1905.11255/) (2019)
- [The linear conditional expectation in Hilbert space](https://mlsanity.com/paper/2008.12070/) (2020)
- [Markov Chain Importance Sampling -- a highly efficient estimator for MCMC](https://mlsanity.com/paper/1805.07179/) (2018)
- [Transporting Higher-Order Quadrature Rules: Quasi-Monte Carlo Points and Sparse Grids for Mixture Distributions](https://mlsanity.com/paper/2308.10081/) (2023)

## Abstract (arXiv)

The Fokker-Planck equation can be reformulated as a continuity equation, which naturally suggests using the associated velocity field in particle flow methods. While the resulting probability flow ODE offers appealing properties - such as defining a gradient flow of the Kullback-Leibler divergence between the current and target densities with respect to the 2-Wasserstein distance - it relies on evaluating the current probability density, which is intractable in most practical applications. By closely examining the drawbacks of approximating this density via kernel density estimation, we uncover opportunities to turn these limitations into advantages in contexts such as variational inference, kernel mean embeddings, and sequential Monte Carlo.