# On the Inverse Flow Matching Problem in the One-Dimensional and Gaussian Cases

arXiv: https://arxiv.org/abs/2512.23265 · HTML: https://mlsanity.com/paper/2512.23265/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 3 of its 5 references that are in the map.
- First posted on arXiv 29 Dec 2025.

## 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, 3 of 3)

- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Flow Generator Matching](https://mlsanity.com/paper/2410.19310/) (2024)

## Abstract (arXiv)

This paper studies the inverse problem of flow matching (FM) between distributions with finite exponential moment, a problem motivated by modern generative AI applications such as the distillation of flow matching models. Uniqueness of the solution is established in two cases - the one-dimensional setting and the Gaussian case. The general multidimensional problem remains open for future studies.