# Verlet Flows: Exact-Likelihood Integrators for Flow-Based Generative Models

arXiv: https://arxiv.org/abs/2405.02805 · HTML: https://mlsanity.com/paper/2405.02805/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 12 references that are in the map.
- First posted on arXiv 5 May 2024.

## Builds on (most cited first, 10 of 10)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [SE(3) Equivariant Augmented Coupling Flows](https://mlsanity.com/paper/2308.10364/) (2023)
- [Scalable Normalizing Flows Enable Boltzmann Generators for Macromolecules](https://mlsanity.com/paper/2401.04246/) (2024)

## Built on by (most cited first, 2 of 2)

- [Flow Perturbation to Accelerate Unbiased Sampling of Boltzmann distribution](https://mlsanity.com/paper/2407.10666/) (2024)
- [Flow Perturbation++: Multi-Step Unbiased Jacobian Estimation for High-Dimensional Boltzmann Sampling](https://mlsanity.com/paper/2601.21177/) (2026)

## Abstract (arXiv)

Approximations in computing model likelihoods with continuous normalizing flows (CNFs) hinder the use of these models for importance sampling of Boltzmann distributions, where exact likelihoods are required. In this work, we present Verlet flows, a class of CNFs on an augmented state-space inspired by symplectic integrators from Hamiltonian dynamics. When used with carefully constructed Taylor-Verlet integrators, Verlet flows provide exact-likelihood generative models which generalize coupled flow architectures from a non-continuous setting while imposing minimal expressivity constraints. On experiments over toy densities, we demonstrate that the variance of the commonly used Hutchinson trace estimator is unsuitable for importance sampling, whereas Verlet flows perform comparably to full autograd trace computations while being significantly faster.