D-Flow SGLD: Source-Space Posterior Sampling for Scientific Inverse Problems with Flow Matching

TL;DR

This paper introduces D-Flow SGLD, a scalable, training-free posterior sampling method for scientific inverse problems using Flow Matching priors, enabling uncertainty quantification without retraining or modifying the learned dynamics.

Contribution

The paper proposes D-Flow SGLD, a novel source-space posterior sampling technique that combines differentiable source inference with stochastic gradient Langevin dynamics for flow matching models.

Findings

D-Flow SGLD effectively explores source posteriors across diverse scientific problems.

It balances measurement assimilation, posterior diversity, and fidelity to physics.

Benchmark results demonstrate its practicality and accuracy in complex inverse problems.

Abstract

Data assimilation and scientific inverse problems require reconstructing high-dimensional physical states from sparse and noisy observations, ideally with uncertainty-aware posterior samples that remain faithful to learned priors and governing physics. While training-free conditional generation is well developed for diffusion models, corresponding conditioning and posterior sampling strategies for Flow Matching (FM) priors remain comparatively under-explored, especially on scientific benchmarks where fidelity must be assessed beyond measurement misfit. In this work, we study training-free conditional generation for scientific inverse problems under FM priors and organize existing inference-time strategies by where measurement information is injected: (i) guided transport dynamics that perturb sampling trajectories using likelihood information, and (ii) source-distribution inference that performs posterior inference over the source variable while keeping the learned transport fixed. Building on the latter, we propose D-Flow SGLD, a source-space posterior sampling method that augments differentiable source inference with preconditioned stochastic gradient Langevin dynamics, enabling scalable exploration of the source posterior induced by new measurement operators without retraining the prior or modifying the learned FM dynamics. We benchmark representative methods from both families on a hierarchy of problems: 2D toy posteriors, chaotic Kuramoto-Sivashinsky trajectories, and wall-bounded turbulence reconstruction. Across these settings, we quantify trade-offs among measurement assimilation, posterior diversity, and physics/statistics fidelity, and establish D-Flow SGLD as a practical FM-compatible posterior sampler for scientific inverse problems.