# Flow Matching for Fast Posterior Sampling in Bayesian Inverse Problems

arXiv: https://arxiv.org/abs/2610.02377 · HTML: https://mlsanity.com/paper/2610.02377/

## 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 22 of its 52 references that are in the map.
- First posted on arXiv 1 Oct 2026.

## 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, 22 of 22)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [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)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Flow Matching for Scalable Simulation-Based Inference](https://mlsanity.com/paper/2305.17161/) (2023)
- [Error Bounds for Flow Matching Methods](https://mlsanity.com/paper/2305.16860/) (2023)
- [Conditional Sampling with Monotone GANs: from Generative Models to Likelihood-Free Inference](https://mlsanity.com/paper/2006.06755/) (2020)
- [Conditional score-based diffusion models for Bayesian inference in infinite dimensions](https://mlsanity.com/paper/2305.19147/) (2023)
- [Dimension-independent likelihood-informed MCMC](https://mlsanity.com/paper/1411.3688/) (2014)
- [Conditional Wasserstein Distances with Applications in Bayesian OT Flow Matching](https://mlsanity.com/paper/2403.18705/) (2024)
- [On Sampling with Approximate Transport Maps](https://mlsanity.com/paper/2302.04763/) (2023)
- [Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inference](https://mlsanity.com/paper/2310.16975/) (2023)
- [Flow Matching: Markov Kernels, Stochastic Processes and Transport Plans](https://mlsanity.com/paper/2501.16839/) (2025)
- [Conditional Generative Models are Provably Robust: Pointwise Guarantees for Bayesian Inverse Problems](https://mlsanity.com/paper/2303.15845/) (2023)
- [Iterative Updating of Model Error for Bayesian Inversion](https://mlsanity.com/paper/1707.04246/) (2017)
- [Hierarchical Bayesian Level Set Inversion](https://mlsanity.com/paper/1601.03605/) (2016)
- [Metropolis-Hastings view on variational inference and adversarial training](https://mlsanity.com/paper/1810.07151/) (2018)
- [How well do generative models solve inverse problems? A benchmark study](https://mlsanity.com/paper/2601.23238/) (2026)

## Abstract (arXiv)

Sampling from the posterior is the central task of computational Bayesian inverse problems. The standard workhorse in Bayesian inference - Markov chain Monte Carlo (MCMC) - is sequential, yields correlated samples, and must be rerun for each observation. Conditional flow matching offers an amortized alternative: a transport map, trained once on joint samples of parameter and data, that yields independent approximate posterior samples for any observation at negligible online cost, without new likelihood evaluations. We give a careful, MCMC-literate assessment of flow matching for PDE-based inverse problems with function-valued parameters. Exploiting the flow's tractable density, we derive computable accuracy estimates of the underlying approximate posterior in total-variation distance and Kullback-Leibler divergence and, moreover, propose a hybrid sampler that is asymptotically exact by Metropolization. We validate the accuracy estimates and demonstrate the amortization in several numerical examples, including electrical impedance tomography and a likelihood-free Lotka-Volterra model.