# Explicit Flow Matching: On The Theory of Flow Matching Algorithms with Applications

arXiv: https://arxiv.org/abs/2402.03232 · HTML: https://mlsanity.com/paper/2402.03232/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 16 of its 21 references that are in the map.
- First posted on arXiv 5 Feb 2024.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 16 of 16)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [Simulation-free Schr"odinger bridges via score and flow matching](https://mlsanity.com/paper/2307.03672/) (2023)
- [On the Generalization Properties of Diffusion Models](https://mlsanity.com/paper/2311.01797/) (2023)
- [Generating and Imputing Tabular Data via Diffusion and Flow-based Gradient-Boosted Trees](https://mlsanity.com/paper/2309.09968/) (2023)
- [Learning Mixtures of Gaussians Using the DDPM Objective](https://mlsanity.com/paper/2307.01178/) (2023)
- [BR-SNIS: Bias Reduced Self-Normalized Importance Sampling](https://mlsanity.com/paper/2207.06364/) (2022)

## Built on by (most cited first, 6 of 6)

- [On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticity](https://mlsanity.com/paper/2506.03719/) (2025)
- [Operator Flow Matching for Timeseries Forecasting](https://mlsanity.com/paper/2510.15101/) (2025)
- [Categorical Reparameterization with Denoising Diffusion models](https://mlsanity.com/paper/2601.00781/) (2026)
- [Unlocking the Duality between Flow and Field Matching](https://mlsanity.com/paper/2602.02261/) (2026)
- [Electric Currents for Discrete Data Generation](https://mlsanity.com/paper/2509.23825/) (2025)
- [Demystifying Transition Matching: When and Why It Can Beat Flow Matching](https://mlsanity.com/paper/2510.17991/) (2025)

## Abstract (arXiv)

This paper proposes a novel method, Explicit Flow Matching (ExFM), for training and analyzing flow-based generative models. ExFM leverages a theoretically grounded loss function, ExFM loss (a tractable form of Flow Matching (FM) loss), to demonstrably reduce variance during training, leading to faster convergence and more stable learning. Based on theoretical analysis of these formulas, we derived exact expressions for the vector field (and score in stochastic cases) for model examples (in particular, for separating multiple exponents), and in some simple cases, exact solutions for trajectories. In addition, we also investigated simple cases of diffusion generative models by adding a stochastic term and obtained an explicit form of the expression for score. While the paper emphasizes the theoretical underpinnings of ExFM, it also showcases its effectiveness through numerical experiments on various datasets, including high-dimensional ones. Compared to traditional FM methods, ExFM achieves superior performance in terms of both learning speed and final outcomes.