Decoupled Diffusion Sampling for Inverse Problems on Function Spaces

TL;DR

This paper introduces a decoupled diffusion framework for inverse PDE problems in function space, improving data efficiency and physics-informed learning while avoiding over-smoothing, with theoretical guarantees and state-of-the-art empirical results.

Contribution

It proposes a novel decoupled diffusion approach that separately models coefficients and PDE guidance, enhancing performance and theoretical robustness in inverse problems.

Findings

Achieves 11% lower $l_2$ error and 54% lower spectral error on average.

Maintains accuracy with 40% better $l_2$ error at 1% data.

Supports effective physics-informed learning and avoids guidance attenuation.

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.