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

2026-02-24

AI Summary

This work addresses posterior reconstruction of high-dimensional physical states in scientific inverse problems under sparse, noisy observations, where fidelity to physical laws and uncertainty quantification must be jointly preserved. Existing flow matching models lack training-free conditional generation and efficient posterior sampling mechanisms. We propose D-Flow SGLD, a differentiable posterior inference framework that integrates preconditioned stochastic gradient Langevin dynamics (SGLD) in the source space, enabling flexible adaptation to arbitrary observation operators without retraining the prior or altering the flow dynamics. We systematically categorize two inference strategies for injecting measurement information in flow matching and establish a scalable, training-free posterior sampling framework. Experiments on 2D toy distributions, Kuramoto–Sivashinsky chaotic trajectories, and wall turbulence reconstruction demonstrate superior balance among measurement consistency, posterior diversity, and physical/statistical fidelity.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Security and Privacy: Large-scale security measurements

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.

Problem

• scientific inverse problems
• data assimilation
• posterior sampling
• flow matching
• uncertainty quantification

Innovation

• Flow Matching
• Posterior Sampling
• Source-Space Inference
• Stochastic Gradient Langevin Dynamics
• Scientific Inverse Problems