# Bayesian Inverse Problems Meet Flow Matching: Efficient and Flexible Inference via Transformers

arXiv: https://arxiv.org/abs/2503.01375 · HTML: https://mlsanity.com/paper/2503.01375/

## Key facts

- Cited by 1 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 13 of its 21 references that are in the map.
- First posted on arXiv 3 Mar 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, 13 of 13)

- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [A Probabilistic Graphical Model Foundation for Enabling Predictive Digital Twins at Scale](https://mlsanity.com/paper/2012.05841/) (2020)
- [Composing Normalizing Flows for Inverse Problems](https://mlsanity.com/paper/2002.11743/) (2020)
- [Solving Inverse Problems with Score-Based Generative Priors learned from Noisy Data](https://mlsanity.com/paper/2305.01166/) (2023)
- [Solving Bayesian Inverse Problems via Variational Autoencoders](https://mlsanity.com/paper/1912.04212/) (2019)
- [Bayesian multiscale deep generative model for the solution of high-dimensional inverse problems](https://mlsanity.com/paper/2102.03169/) (2021)
- [Tractable Optimal Experimental Design using Transport Maps](https://mlsanity.com/paper/2401.07971/) (2024)
- [Efficient Bayesian inference using physics-informed invertible neural networks for inverse problems](https://mlsanity.com/paper/2304.12541/) (2023)
- [ContextFlow++: Generalist-Specialist Flow-based Generative Models with Mixed-Variable Context Encoding](https://mlsanity.com/paper/2406.00578/) (2024)

## Built on by (most cited first, 1 of 1)

- [OCL-PDE: A Generative Framework for PDE Inverse Problems with Observation-Complementary Latents](https://mlsanity.com/paper/2610.06259/) (2026)

## Abstract (arXiv)

The efficient resolution of Bayesian inverse problems remains challenging due to the high computational cost of traditional sampling methods. In this paper, we propose a novel framework that integrates Conditional Flow Matching (CFM) with a transformer-based architecture to enable fast and flexible sampling from complex posterior distributions. The proposed methodology involves the direct learning of conditional probability trajectories from the data, leveraging CFM's ability to bypass iterative simulation and transformers' capacity to process arbitrary numbers of observations. The efficacy of the proposed framework is demonstrated through its application to three problems: a simple nonlinear model, a disease dynamics framework, and a two-dimensional Darcy flow Partial Differential Equation. The primary outcomes demonstrate that the relative errors in parameters recovery are as low as 1.5%, and that the inference time is reduced by up to 2000 times on CPU in comparison with the Monte Carlo Markov Chain. This framework facilitates the expeditious resolution of Bayesian problems through the utilisation of sampling from the learned conditional distribution.