FlowSGS introduces a flow-based posterior sampling method that combines Split Gibbs Sampling (SGS) with Langevin dynamics for the likelihood step and Stochastic Interpolants (SI) for the prior step. By integrating a pretrained flow model into the prior step via SI's reverse-time SDE and a novel timestep correction, FlowSGS reduces the number of network evaluations compared to plug‑and‑play diffusion samplers. Experiments demonstrate state‑of‑the‑art performance on various inverse problems, including the first flow‑based solution to a nonlinear inverse problem (Fourier phase retrieval).
By Tianao Li, Xinhui Qian, Emma Alexander
The paper introduces flow matching as an amortized alternative to Markov chain Monte Carlo for sampling from posterior distributions in Bayesian inverse problems. By training a transport map on joint samples of parameters and data, the method produces independent approximate posterior samples for any observation at negligible online cost, eliminating the need for new likelihood evaluations. The authors provide accuracy estimates in total‑variation distance and Kullback‑Leibler divergence, propose a hybrid Metropolized sampler for exactness, and validate the approach on examples such as electrical impedance tomography and a likelihood‑free Lotka‑Volterra model.
By Jan Blechschmidt, Oliver G. Ernst, Moritz Poguntke, Bj\"orn Sprungk
The paper introduces MS-Flow, a method that represents a flow-based generative model’s trajectory as a sequence of intermediate latent states instead of a single initial code. By enforcing local flow dynamics and coupling trajectory segments with matching penalties, the approach alternates between updating latent states and ensuring consistency with observed data. This strategy reduces memory usage and improves reconstruction quality on tasks such as image inpainting, super‑resolution, and computed tomography.
By Alexander Denker, Zeljko Kereta, Carola-Bibiane Sch\"onlieb, Moshe Eliasof
The paper introduces OCL-PDE, a generative framework for solving partial differential equation (PDE) inverse problems. It uses a learned observation‑complementary latent representation that retains reconstruction‑relevant information and combines it with observations to recover unknown fields. The method, built on a physics‑aware autoencoder and conditional Flow Matching, improves reconstruction accuracy and fine‑detail recovery compared to baseline approaches.
arXiv:2610. 09281v1 Announce Type: new Abstract: In Bayesian inverse problems, posterior sampling requires generating samples that are consistent with given observations while capturing the range of plausible solutions.
By Shiqin Zeng, Zijun Deng, Felix J. Herrmann
PosteriorBench is a new benchmark that evaluates how well generative inverse solvers recover full posterior distributions rather than just a single reconstruction. It tests four physics-based inverse problems—Darcy flow inversion, Poisson source recovery, carbon capture and storage, and light transport material inference—using high-fidelity reference posteriors generated by rejection sampling and MCMC. The benchmark employs five metrics (posterior-mean error, posterior-standard-deviation error, maximum mean discrepancy, sliced Wasserstein distance, and radially averaged power-spectrum error) to assess pointwise accuracy, uncertainty, distributional alignment, and global frequency fidelity, revealing significant distribution-matching gaps in current solvers and highlighting the importance of neural operators, guidance weights, and generation noise for posterior-variance calibration.
By Jiachen Yao, Zi-Siang Hsu, Xi Deng, Aditi Gupta, Xin Ju, Sally M Benson, Gege Wen, Anima Anandkumar