# What Do Flow-Based Inverse Solvers Approximate? A Posterior-Transport View

arXiv: https://arxiv.org/abs/2606.24516 · HTML: https://mlsanity.com/paper/2606.24516/

## 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 24 of its 30 references that are in the map.
- First posted on arXiv 23 Jun 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, 24 of 24)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model](https://mlsanity.com/paper/2212.00490/) (2022)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [Practical and Asymptotically Exact Conditional Sampling in Diffusion Models](https://mlsanity.com/paper/2306.17775/) (2023)
- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing](https://mlsanity.com/paper/2407.01521/) (2024)
- [Score-Based Diffusion Models as Principled Priors for Inverse Imaging](https://mlsanity.com/paper/2304.11751/) (2023)
- [D-Flow: Differentiating through Flows for Controlled Generation](https://mlsanity.com/paper/2402.14017/) (2024)
- [Theoretical guarantees for sampling and inference in generative models with latent diffusions](https://mlsanity.com/paper/1903.01608/) (2019)
- [PnP-Flow: Plug-and-Play Image Restoration with Flow Matching](https://mlsanity.com/paper/2410.02423/) (2024)
- [Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstruction](https://mlsanity.com/paper/2403.17042/) (2024)
- [FlowDPS: Flow-Driven Posterior Sampling for Inverse Problems](https://mlsanity.com/paper/2503.08136/) (2025)
- [DEFT: Efficient Fine-Tuning of Diffusion Models by Learning the Generalised h-transform](https://mlsanity.com/paper/2406.01781/) (2024)
- [Variational Diffusion Posterior Sampling with Midpoint Guidance](https://mlsanity.com/paper/2410.09945/) (2024)
- [Flower: A Flow-Matching Solver for Inverse Problems](https://mlsanity.com/paper/2509.26287/) (2025)
- [SITCOM: Step-wise Triple-Consistent Diffusion Sampling for Inverse Problems](https://mlsanity.com/paper/2410.04479/) (2024)
- [Action-Minimization Meets Generative Modeling: Efficient Transition Path Sampling with the Onsager-Machlup Functional](https://mlsanity.com/paper/2504.18506/) (2025)
- [Onsager-Machlup Posterior Transport for Deep Gaussian Processes](https://mlsanity.com/paper/2605.23434/) (2026)

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

- [Fast and Faithful: Principled Conditional Flow Matching for Inverse Problems](https://mlsanity.com/paper/2609.12953/) (2026)

## Abstract (arXiv)

A growing family of training-free solvers -- FlowDPS, FLOWER, PnP-Flow and their diffusion ancestors (DPS, DAPS) -- repurpose a pretrained flow-matching prior to solve imaging inverse problems by adding a measurement-guidance term to the deterministic probability-flow ODE. Despite strong empirical results, what these per-step corrections actually approximate -- and how far the resulting samples are from the true posterior p(xmid y) -- has not been characterized. We give a posterior-transport account of flow-based inverse problem solving. Our starting point is a simple but consequential fact: for a deterministic flow prior, Bayesian conditioning is realized entirely by a reweighting of the source distribution, not by a drift correction; pushing the reweighted source through the unmodified velocity field yields exact posterior samples. From this we show that trajectory-guidance solvers can be read as the minimum-kinetic-energy correction field needed to morph the unconditional source into the posterior, and that FlowDPS / FLOWER / PnP-Flow correspond to distinct zeroth-order / Gaussian / proximal approximations of this single object; we bound the resulting posterior bias in Wasserstein distance. A controlled 2D study with a closed-form posterior confirms the theory decisively: source reweighting matches the true posterior to the Monte-Carlo floor on every metric, whereas trajectory guidance incurs 200--800× larger error and collapses posterior modes, regardless of guidance strength. Guided by the analysis we propose a cheap, principled velocity-correction solver that is competitive across two in-domain priors (AFHQ, CelebA) and two out-of-distribution settings while, unlike point-estimate source-space optimizers, producing diverse posterior samples with uncertainty that correlates with reconstruction error.