D-Flow SGLD: Source-Space Posterior Sampling for Scientific Inverse Problems with Flow Matching
arXiv:2602.21469v1 Announce Type: new 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 variab
arXiv:2602.21469v1 Announce Type: new 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.
Executive Summary
This article presents D-Flow SGLD, a novel source-space posterior sampling method for scientific inverse problems under Flow Matching (FM) priors. D-Flow SGLD augments differentiable source inference with preconditioned stochastic gradient Langevin dynamics, enabling scalable exploration of the source posterior without retraining the prior or modifying the learned FM dynamics. The authors benchmark D-Flow SGLD against existing methods on a hierarchy of problems, demonstrating its potential as a practical FM-compatible posterior sampler. The work addresses a significant gap in the literature, where FM priors have been under-explored for scientific inverse problems. The authors' approach provides a scalable and uncertainty-aware solution for reconstructing high-dimensional physical states from sparse and noisy observations.
Key Points
- ▸ D-Flow SGLD is a novel source-space posterior sampling method for scientific inverse problems under FM priors.
- ▸ D-Flow SGLD augments differentiable source inference with preconditioned stochastic gradient Langevin dynamics.
- ▸ D-Flow SGLD enables scalable exploration of the source posterior without retraining the prior or modifying the learned FM dynamics.
Merits
Strength in Uncertainty-Awareness
D-Flow SGLD provides uncertainty-aware posterior samples that remain faithful to learned priors and governing physics, addressing a significant gap in the literature.
Scalability and Efficiency
D-Flow SGLD enables scalable exploration of the source posterior, making it a practical solution for large-scale scientific inverse problems.
Demerits
Limited Evaluation on Complex Domains
While the authors benchmark D-Flow SGLD on a hierarchy of problems, its performance on more complex and realistic domains remains to be explored.
Dependency on FM Priors
D-Flow SGLD is specifically designed for FM priors, limiting its applicability to other types of priors or models.
Expert Commentary
The article presents a novel and significant contribution to the field of Bayesian inference for scientific inverse problems. D-Flow SGLD addresses a critical gap in the literature by providing a scalable and uncertainty-aware solution for reconstructing high-dimensional physical states from sparse and noisy observations. The authors' approach is well-motivated and theoretically sound, and its evaluation on a hierarchy of problems provides valuable insights into its performance. However, further research is needed to explore the limitations and potential applications of D-Flow SGLD, particularly on more complex and realistic domains.
Recommendations
- ✓ To further develop D-Flow SGLD, the authors should explore its application on more complex and realistic scientific inverse problems.
- ✓ Future research should investigate the potential extensions of D-Flow SGLD to other types of priors or models, to broaden its applicability.
