# Learning Optimal Flows for Non-Equilibrium Importance Sampling

arXiv: https://arxiv.org/abs/2206.09908 · HTML: https://mlsanity.com/paper/2206.09908/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 45 references that are in the map.
- First posted on arXiv 20 Jun 2022.

## Builds on (most cited first, 9 of 9)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Painless Stochastic Gradient: Interpolation, Line-Search, and Convergence Rates](https://mlsanity.com/paper/1905.09997/) (2019)
- [Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2102.07501/) (2021)
- [Targeted free energy estimation via learned mappings](https://mlsanity.com/paper/2002.04913/) (2020)
- [A Dacorogna-Moser approach to flow decomposition and minimal flow problems](https://mlsanity.com/paper/1310.2738/) (2013)

## Built on by (most cited first, 4 of 4)

- [A Mathematical Framework for Learning Probability Distributions](https://mlsanity.com/paper/2212.11481/) (2022)
- [Ensemble-Based Annealed Importance Sampling](https://mlsanity.com/paper/2401.15645/) (2024)
- [Beckmann Transport Models: From Autonomous Flows to One-Step Maps](https://mlsanity.com/paper/2608.01692/) (2026)
- [FluxLite: Inference-Time Proposal Control for Discrete Diffusion Models](https://mlsanity.com/paper/2609.35947/) (2026)

## Abstract (arXiv)

Many applications in computational sciences and statistical inference require the computation of expectations with respect to complex high-dimensional distributions with unknown normalization constants, as well as the estimation of these constants. Here we develop a method to perform these calculations based on generating samples from a simple base distribution, transporting them by the flow generated by a velocity field, and performing averages along these flowlines. This non-equilibrium importance sampling (NEIS) strategy is straightforward to implement and can be used for calculations with arbitrary target distributions. On the theory side, we discuss how to tailor the velocity field to the target and establish general conditions under which the proposed estimator is a perfect estimator with zero-variance. We also draw connections between NEIS and approaches based on mapping a base distribution onto a target via a transport map. On the computational side, we show how to use deep learning to represent the velocity field by a neural network and train it towards the zero variance optimum. These results are illustrated numerically on benchmark examples (with dimension up to 10), where after training the velocity field, the variance of the NEIS estimator is reduced by up to 6 orders of magnitude than that of a vanilla estimator. We also compare the performances of NEIS with those of Neal's annealed importance sampling (AIS).