Decoupled Diffusion Sampling for Inverse Problems on Function Spaces

2026-01-30

AI Summary

This work addresses the challenges of heavy reliance on large paired supervised datasets and posterior sampling over-smoothing in inverse problems for partial differential equations (PDEs). To this end, we propose the Decoupled Diffusion Inverse Solver (DDIS), which leverages an unconditional diffusion model to learn the prior over coefficients and explicitly models the forward PDE using a neural operator, thereby enabling physics-informed, efficient inversion. Our method introduces a decoupled architecture that mitigates guidance decay under data scarcity by avoiding joint modeling, and incorporates a novel Decoupled Annealed Posterior Sampling (DAPS) strategy to effectively suppress over-smoothing. Experiments demonstrate that with only 1% of the training data, DDIS reduces the L2 error by 40% compared to joint models; in sparse observation settings, it achieves average reductions of 11% in L2 error and 54% in spectral error.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We propose a data-efficient, physics-aware generative framework in function space for inverse PDE problems. Existing plug-and-play diffusion posterior samplers represent physics implicitly through joint coefficient-solution modeling, requiring substantial paired supervision. In contrast, our Decoupled Diffusion Inverse Solver (DDIS) employs a decoupled design: an unconditional diffusion learns the coefficient prior, while a neural operator explicitly models the forward PDE for guidance. This decoupling enables superior data efficiency and effective physics-informed learning, while naturally supporting Decoupled Annealing Posterior Sampling (DAPS) to avoid over-smoothing in Diffusion Posterior Sampling (DPS). Theoretically, we prove that DDIS avoids the guidance attenuation failure of joint models when training data is scarce. Empirically, DDIS achieves state-of-the-art performance under sparse observation, improving $l_2$ error by 11% and spectral error by 54% on average; when data is limited to 1%, DDIS maintains accuracy with 40% advantage in $l_2$ error compared to joint models.

Problem

• inverse problems
• function spaces
• diffusion models
• PDE
• data efficiency

Innovation

• Decoupled Diffusion
• Physics-Informed Learning
• Neural Operator
• Inverse PDE Problems
• Posterior Sampling