# Convergence of Continuous Normalizing Flows for Learning Probability Distributions

arXiv: https://arxiv.org/abs/2404.00551 · HTML: https://mlsanity.com/paper/2404.00551/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 100 references that are in the map.
- First posted on arXiv 31 Mar 2024.

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

- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [Wasserstein Convergence Guarantees for a General Class of Score-Based Generative Models](https://mlsanity.com/paper/2311.11003/) (2023)
- [Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distances](https://mlsanity.com/paper/2401.17958/) (2024)
- [Analysis of learning a flow-based generative model from limited sample complexity](https://mlsanity.com/paper/2310.03575/) (2023)
- [Convergence of flow-based generative models via proximal gradient descent in Wasserstein space](https://mlsanity.com/paper/2310.17582/) (2023)
- [Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivatives](https://mlsanity.com/paper/2305.08466/) (2023)
- [On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimates](https://mlsanity.com/paper/2311.13584/) (2023)
- [Improved Convergence of Score-Based Diffusion Models via Prediction-Correction](https://mlsanity.com/paper/2305.14164/) (2023)
- [Distribution Approximation and Statistical Estimation Guarantees of Generative Adversarial Networks](https://mlsanity.com/paper/2002.03938/) (2020)
- [Distribution learning via neural differential equations: a nonparametric statistical perspective](https://mlsanity.com/paper/2309.01043/) (2023)
- [Gaussian Interpolation Flows](https://mlsanity.com/paper/2311.11475/) (2023)
- [Conditional Stochastic Interpolation for Generative Learning](https://mlsanity.com/paper/2312.05579/) (2023)
- [Deep conditional distribution learning via conditional F"ollmer flow](https://mlsanity.com/paper/2402.01460/) (2024)
- [Convergence of the Inexact Langevin Algorithm in KL Divergence with Application to Score-based Generative Models](https://mlsanity.com/paper/2211.01512/) (2022)
- [Estimation of Non-Crossing Quantile Regression Process with Deep ReQU Neural Networks](https://mlsanity.com/paper/2207.10442/) (2022)

## Built on by (most cited first, 26 of 26)

- [A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models](https://mlsanity.com/paper/2408.02320/) (2024)
- [Learning stochastic dynamics from snapshots through regularized unbalanced optimal transport](https://mlsanity.com/paper/2410.00844/) (2024)
- [Theoretical guarantees in KL for Diffusion Flow Matching](https://mlsanity.com/paper/2409.08311/) (2024)
- [Latent Schr"odinger Bridge Diffusion Model for Generative Learning](https://mlsanity.com/paper/2404.13309/) (2024)
- [On the minimax optimality of Flow Matching through the connection to kernel density estimation](https://mlsanity.com/paper/2504.13336/) (2025)
- [Flow Matching is Adaptive to Manifold Structures](https://mlsanity.com/paper/2602.22486/) (2026)
- [Integrating Dynamical Systems Modeling with Spatiotemporal scRNA-seq Data Analysis](https://mlsanity.com/paper/2503.11347/) (2025)
- [Error Analysis of Discrete Flow with Generator Matching](https://mlsanity.com/paper/2509.21906/) (2025)
- [Lipschitz regularity in Flow Matching and Diffusion Models: sharp sampling rates and functional inequalities](https://mlsanity.com/paper/2604.06065/) (2026)
- [Theory on Score-Mismatched Diffusion Models and Zero-Shot Conditional Samplers](https://mlsanity.com/paper/2410.13746/) (2024)
- [Chance-constrained Flow Matching for High-Fidelity Constraint-aware Generation](https://mlsanity.com/paper/2509.25157/) (2025)
- [Distribution estimation via Flow Matching with Lipschitz guarantees](https://mlsanity.com/paper/2509.02337/) (2025)
- [Local Flow Matching Generative Models](https://mlsanity.com/paper/2410.02548/) (2024)
- [Pathway to O(√d) Complexity bound under Wasserstein metric of flow-based models](https://mlsanity.com/paper/2512.06702/) (2025)
- [Model Free Prediction with Uncertainty Assessment](https://mlsanity.com/paper/2405.12684/) (2024)
- [Elucidating Rectified Flow with Deterministic Sampler: Polynomial Discretization Complexity for Multi and One-step Models](https://mlsanity.com/paper/2508.08735/) (2025)
- [On Conditional Stochastic Interpolation for Generative Nonlinear Sufficient Dimension Reduction](https://mlsanity.com/paper/2512.18971/) (2025)
- [From Conditional to Unconditional Independence: Testing Conditional Independence via Transport Maps](https://mlsanity.com/paper/2504.09567/) (2025)
- [Graph Flow Matching: Enhancing Image Generation with Neighbor-Aware Flow Fields](https://mlsanity.com/paper/2505.24434/) (2025)
- [Non-asymptotic convergence bound of conditional diffusion models](https://mlsanity.com/paper/2508.10944/) (2025)
- [Max-Entropy Reinforcement Learning with Flow Matching and A Case Study on LQR](https://mlsanity.com/paper/2512.23870/) (2025)
- [Generalization Bounds for Flow-matching Generative Models for Intrinsically Low-dimensional Data](https://mlsanity.com/paper/2610.02663/) (2026)
- [Smooth Flow Matching for Synthesizing Functional Data](https://mlsanity.com/paper/2508.13831/) (2025)
- [RepFlow: Representation Enhanced Flow Matching for Causal Effect Estimation](https://mlsanity.com/paper/2605.05890/) (2026)
- [Ergodic Trajectory Design by Learned Pushforward Maps: Provable Coverage via Conditional Flow Matching](https://mlsanity.com/paper/2605.13063/) (2026)
- [When do explicit penalties improve flow matching?](https://mlsanity.com/paper/2610.15689/) (2026)

## Abstract (arXiv)

Continuous normalizing flows (CNFs) are a generative method for learning probability distributions, which is based on ordinary differential equations. This method has shown remarkable empirical success across various applications, including large-scale image synthesis, protein structure prediction, and molecule generation. In this work, we study the theoretical properties of CNFs with linear interpolation in learning probability distributions from a finite random sample, using a flow matching objective function. We establish non-asymptotic error bounds for the distribution estimator based on CNFs, in terms of the Wasserstein-2 distance. The key assumption in our analysis is that the target distribution satisfies one of the following three conditions: it either has a bounded support, is strongly log-concave, or is a finite or infinite mixture of Gaussian distributions. We present a convergence analysis framework that encompasses the error due to velocity estimation, the discretization error, and the early stopping error. A key step in our analysis involves establishing the regularity properties of the velocity field and its estimator for CNFs constructed with linear interpolation. This necessitates the development of uniform error bounds with Lipschitz regularity control of deep ReLU networks that approximate the Lipschitz function class, which could be of independent interest. Our nonparametric convergence analysis offers theoretical guarantees for using CNFs to learn probability distributions from a finite random sample.