# Flower: A Flow-Matching Solver for Inverse Problems

arXiv: https://arxiv.org/abs/2509.26287 · HTML: https://mlsanity.com/paper/2509.26287/

## Key facts

- Cited by 27 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 33 of its 43 references that are in the map.
- First posted on arXiv 30 Sep 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, 30 of 33)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [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)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [ILVR: Conditioning Method for Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2108.02938/) (2021)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Plug-and-Play Image Restoration with Deep Denoiser Prior](https://mlsanity.com/paper/2008.13751/) (2020)
- [Denoising Diffusion Models for Plug-and-Play Image Restoration](https://mlsanity.com/paper/2305.08995/) (2023)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [A Variational Perspective on Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2305.04391/) (2023)
- [Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing](https://mlsanity.com/paper/2407.01521/) (2024)
- [Decomposed Diffusion Sampler for Accelerating Large-Scale Inverse Problems](https://mlsanity.com/paper/2303.05754/) (2023)
- [D-Flow: Differentiating through Flows for Controlled Generation](https://mlsanity.com/paper/2402.14017/) (2024)
- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [PnP-Flow: Plug-and-Play Image Restoration with Flow Matching](https://mlsanity.com/paper/2410.02423/) (2024)
- [Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularization](https://mlsanity.com/paper/2201.13256/) (2022)
- [Training-free Linear Image Inverses via Flows](https://mlsanity.com/paper/2310.04432/) (2023)
- [FlowDPS: Flow-Driven Posterior Sampling for Inverse Problems](https://mlsanity.com/paper/2503.08136/) (2025)
- [Flow Priors for Linear Inverse Problems via Iterative Corrupted Trajectory Matching](https://mlsanity.com/paper/2405.18816/) (2024)
- [From Facial Parts Responses to Face Detection: A Deep Learning Approach](https://mlsanity.com/paper/1509.06451/) (2015)
- [Stochastic Normalizing Flows for Inverse Problems: a Markov Chains Viewpoint](https://mlsanity.com/paper/2109.11375/) (2021)
- [Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithms](https://mlsanity.com/paper/2308.10542/) (2023)

## Built on by (most cited first, 27 of 27)

- [FlowLPS: Langevin-Proximal Sampling for Flow-based Inverse Problem Solvers](https://mlsanity.com/paper/2512.07150/) (2025)
- [Solving Room Impulse Response Inverse Problems Using Flow Matching with Analytic Wiener Denoiser](https://mlsanity.com/paper/2602.00652/) (2026)
- [Conditional flow matching for physics-constrained inverse problems with finite training data](https://mlsanity.com/paper/2603.14135/) (2026)
- [FluxFlow: Conservative Flow-Matching for Astronomical Image Super-Resolution](https://mlsanity.com/paper/2605.03749/) (2026)
- [A Stability Benchmark of Generative Regularizers for Inverse Problems](https://mlsanity.com/paper/2605.10076/) (2026)
- [NullFlow: One-Step Generative Reconstruction](https://mlsanity.com/paper/2606.22696/) (2026)
- [What Do Flow-Based Inverse Solvers Approximate? A Posterior-Transport View](https://mlsanity.com/paper/2606.24516/) (2026)
- [Physics-Guided Flow Matching for CT Image Reconstruction](https://mlsanity.com/paper/2608.28256/) (2026)
- [Efficient Flow Matching for Sparse-View CT Reconstruction](https://mlsanity.com/paper/2603.00205/) (2026)
- [P-Flow: Proxy-gradient Flows for Linear Inverse Problems](https://mlsanity.com/paper/2605.08328/) (2026)
- [FlowADMM: Plug-and-play ADMM with Flow-based Renoise-Denoise Priors](https://mlsanity.com/paper/2605.08640/) (2026)
- [Triadic Dynamics Aware Diffusion Posterior Sampling for Inverse Problems: Optimizing Guidance and Stochasticity Schedules](https://mlsanity.com/paper/2605.26470/) (2026)
- [A Distributionally Robust Framework for Learned Reconstructions in Inverse Problems](https://mlsanity.com/paper/2606.30230/) (2026)
- [Constrained Flow Matching via Lagrangian Dual Flows](https://mlsanity.com/paper/2607.04513/) (2026)
- [Hybrid-Domain Posterior Sampling for Inverse Problems via Latent Flow Matching](https://mlsanity.com/paper/2608.00537/) (2026)
- [Making Every Step Count: Spatio-Temporal Information Allocation for Imaging Inverse Problems](https://mlsanity.com/paper/2608.11747/) (2026)
- [Flow Matching-Based PET Image Reconstruction](https://mlsanity.com/paper/2608.20112/) (2026)
- [SurfSpec: Enhancing Off-Target-Agnostic Specificity by Bounding Pocket-Ligand Geometric Mismatch](https://mlsanity.com/paper/2609.02963/) (2026)
- [ParetoTransport: Generative Optimization by Mass Transport Toward The Pareto Front](https://mlsanity.com/paper/2609.07706/) (2026)
- [Fast and Faithful: Principled Conditional Flow Matching for Inverse Problems](https://mlsanity.com/paper/2609.12953/) (2026)
- [CLIMB-flow: Coupled Linear Inverse posterior sampling via Multiscale-Based flow](https://mlsanity.com/paper/2609.33834/) (2026)
- [SNaP: One-Step Posterior Sampling for Noisy Inverse Problems](https://mlsanity.com/paper/2609.34071/) (2026)
- [Principled MAP estimation for inverse problems: bridging the gap between convergence and performance](https://mlsanity.com/paper/2609.37529/) (2026)
- [Getting Your Guidance Weights Right in diffusion and flow-matching posterior sampling](https://mlsanity.com/paper/2610.03503/) (2026)
- [Stochastic Finite-Transition Learning and Exploration for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2610.13095/) (2026)
- [Spanning Diffusion and Flow-based Posterior Sampling via Exact Gaussian Conditioning](https://mlsanity.com/paper/2610.13812/) (2026)
- [BridgeTwist: Twisting Schr"odinger Bridges for Training-Free Conditional Sampling](https://mlsanity.com/paper/2610.18658/) (2026)

## Abstract (arXiv)

We introduce Flower, a solver for linear inverse problems. It leverages a pre-trained flow model to produce reconstructions that are consistent with the observed measurements. Flower operates through an iterative procedure over three steps: (i) a flow-consistent destination estimation, where the velocity network predicts a denoised target; (ii) a refinement step that projects the estimated destination onto a feasible set defined by the forward operator; and (iii) a time-progression step that re-projects the refined destination along the flow trajectory. We provide a theoretical analysis that demonstrates how Flower approximates Bayesian posterior sampling, thereby unifying perspectives from plug-and-play methods and generative inverse solvers. On the practical side, Flower achieves state-of-the-art reconstruction quality while using nearly identical hyperparameters across various linear inverse problems. Our code is available at https://github.com/mehrsapo/Flower.